{"id":"8b920e7f-fa52-4846-95b0-7fc43a42b11d","arxiv_id":"2504.19470","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Relative to a particular quantum unitary oracle QMA is not contained in polyQCPH, although QMA is contained in polyQCPH for every classical oracle; the same separation is claimed for a distributional oracle.","lead":"The paper defines polyQCPH, a bounded-error quantum class equal to PSPACE under classical oracles, and shows QMA is contained in it classically but not relative to a specific quantum unitary oracle. This resolves a 2009 open problem of Aaronson and warns that quantum oracle separations can be much weaker than classical ones.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The quantum separation hinges on Eq. (1) in Theorem 4.3, where E[||Φψ-ΦI||^2] ≤ O(T√((m+1)/2^n)) is asserted without derivation; the correct AK07 bound appears to be on the first moment, not the squared distance, and the Markov step must be re-based.","rationale":"Reading in good faith, the central claim is the quantum oracle separation of Theorem 1.1, and the proof of Theorem 4.4 depends on the quantitative distortion bound in Theorem 4.3. The reader's weakest-assumption identification is accurate: Eq. (1) is the load-bearing step. The paper cites AK07 for the bound but states it in a form that is not justified by Lemma 2.13 as presented; the intermediate states in the hybrid expansion are ψ-dependent, and the printed inequality is a second-moment bound with a first-moment-looking right-hand side. The one-query sanity check shows the printed form cannot be the tight estimate used for the Markov argument. However, the underlying AK07 technique is known to yield a first-moment bound of the form E[||Φψ−ΦI||] ≤ O(T√((m+1)/2^n)), from which the diagonalization follows after a routine repair; the theorem statement of Theorem 1.1 is therefore likely correct, but the write-up needs correction. The PQP reduction in Lemma 4.1 is also flawed as noted, but it is not needed for the main separation. The distributional proof has separate gaps in the Markov/union-bound step, but the central quantum claim is what drives the verdict. Thus CONDITIONAL remains the right verdict, and no change from the reader's assessment is warranted.","tokens_in":18874,"tokens_out":27108,"duration_ms":285322,"concrete_test":"Independently re-derive Eq. (1) from Lemma 2.13 following the AK07 hybrid argument, and use a one-query sanity check: for m=0, A starts in |0^n> and applies Uψ once, giving E[||(Uψ−I)|0^n>||^2] = 4/2^n, while the printed RHS of Eq. (1) is O(√(1/2^n)); for n=1 this is 2 vs O(0.707), so the printed squared-distance form is not the tight bound used. Then determine whether the correct bound is E[||Φψ−ΦI||] ≤ O(T√((m+1)/2^n)), and verify that the Markov step in Eq. (2) is applied to that first moment (or to ||Φψ−ΦI||^2 via E[X^2] ≤ 2E[X]) so that the high-probability bound still yields the diagonalization in Theorem 4.4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 4.4 rests entirely on Theorem 4.3, whose proof asserts Eq. (1): E_{|ψ>∼S(w*)}[|| |Φψ> − |ΦI> ||^2] ≤ O(T√((m+1)/2^n)). This is not a direct consequence of Lemma 2.13 as presented. Lemma 2.13 bounds E[<ψ|ρ|ψ>] for a state ρ that is independent of ψ, whereas each intermediate state in the hybrid expansion depends on ψ through earlier queries. Moreover, the printed inequality is a second-moment bound with a first-moment-looking right-hand side: in the standard AK07 hybrid argument, Cauchy–Schwarz and the geometric lemma give E[||Φψ − ΦI||] ≤ O(T√((m+1)/2^n)), not the squared-distance form written here. The subsequent Markov inequality in Eq. (2) is then applied to the squared quantity without a proof that its expectation is small. The one-query sanity check in the concrete test shows the printed form cannot be the tight estimate used. If the intended bound is the first-moment one, the diagonalization in Theorem 4.4 is salvageable by replacing Eq. (1) with the correct statement and adjusting the Markov step; if no such bound can be derived for the p-uniform cell S(w*) against adaptive algorithms, the oracle separation does not follow. A separate, non-central bug is Lemma 4.1: running the QMA verifier on the maximally mixed proof keeps the YES acceptance probability at about 2^{−p(n)}, below the 1/2 threshold required by PQP.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a bounded-error quantum-classical polynomial hierarchy polyQCPH, proves that QMA is contained in polyQCPH both unrelativized and relative to every classical oracle, and claims to construct a quantum unitary oracle U such that QMA^U is not contained in polyQCPH^U. It also claims an analogous separation relative to a distributional oracle. The main conceptual contribution is to exhibit a complexity-class containment that relativizes classically but fails quantumly, thereby resolving an open problem of Aaronson for bounded-error classes. The proofs reuse the Aaronson-Kuperberg quantum oracle and the Li-Liu-Pelecanos-Yamakawa distributional oracle, and introduce polyQCPH as the bounded-error analogue of PSPACE that admits a quantum-oracle interpretation.","tokens_in":19169,"tokens_out":27988,"duration_ms":287373,"significance":"If the main theorem were fully established, it would resolve a question that has been open since 2009 and would provide a concrete cautionary example about the use of quantum and distributional oracle models. The paper is also notable for introducing a natural bounded-error class, polyQCPH, that generalizes QCMA and collapses to PSPACE under classical oracles. The central separation is plausible and the reuse of the AK07 and LLPY24 oracles gives the argument a solid external foundation. However, the manuscript as written contains several load-bearing gaps: the formal definition of polyQCPH does not match the intended polynomial number of alternations, the key geometric bound in Theorem 4.3 is asserted without a supporting derivation, and the proof of Lemma 4.1 is incorrect. These issues prevent acceptance in the current form.","major_comments":[{"comment":"The formal definition of polyQCPH is inconsistent with the intended meaning. Definition 2.9 states that V_n takes proofs y_1,...,y_{n_i}, and Definition 2.11 defines polyQCPH as the union over i∈N of polyQCΣ_i. If the number of proofs is the fixed constant i, the union does not allow the number of alternations to grow with the input length, contradicting the introduction and the use in Theorem 4.4 of q(n_i) alternations. The definitions should be rewritten so that the number of proofs is a polynomial function i(n), and polyQCPH is the union over all polynomial functions i(n). Without this correction, the statement of the main theorem is ambiguous.","section":"Definitions 2.9–2.11"},{"comment":"The bound E_{|ψ>∼S(w*)}[|| |Φψ> − |ΦI> ||^2] ≤ O(T sqrt((m+1)/2^n)) is the pivotal estimate for the quantum oracle separation, but it is asserted without proof. The paper says this follows from Lemma 2.13, yet Lemma 2.13 applies to a density matrix ρ that is independent of ψ, whereas in the hybrid expansion the state before each query depends on ψ through earlier queries. The printed second-moment form is not directly a consequence of the stated lemma. Since Eq. (2) and Theorem 4.4 rest entirely on this bound, the authors must either provide a complete derivation or cite the exact theorem in [AK07] that yields this estimate, explaining how the adaptive dependence of the query algorithm is handled.","section":"Theorem 4.3, Eq. (1)"},{"comment":"The proof of QMA^U⊆PQP^U is incorrect. For a YES instance, feeding the maximally mixed state to a QMA verifier gives the average acceptance probability over all proof states, and the existence of one good proof does not imply a lower bound of 2^{-p(n)}(1−2^{-r(n)}). The average can be arbitrarily small, for example if the verifier accepts only a single computational basis state. The lemma may be true, but this proof does not establish it. The lemma is not needed for the main separation, but as written it is a substantial error in a stated result and should be fixed or removed.","section":"Lemma 4.1"},{"comment":"The claim that L∈QMA^O with probability 1 over the choice of {(b_n,H_n,r_n)} is not justified by Lemma 5.2 item 1. That lemma provides a witness |z_H> that succeeds with probability 1−negl(n) over (H_n,r_n,b_n), but a negligible failure probability for each n does not imply that all n succeed with probability 1, because the failure probabilities may not be summable. The authors should either prove a stronger statement, such as the existence of a witness that works for every r_n up to a measure-zero set, or give the precise theorem in [LLPY24] that establishes L∈QMA^O with probability 1. This is load-bearing for the completeness side of the distributional separation.","section":"Theorem 5.3"}],"minor_comments":[{"comment":"The notation y_{n_i} appears to be a typo; the intended number of proofs should be a polynomial function of n, and the definition should use i(n) rather than the ambiguous subscript n_i.","section":"Definition 2.9"},{"comment":"The proof of PSPACE^O⊆polyPH^O is very terse; it should state explicitly that the TQBF formula constructed in Lemma 3.1 can include oracle predicates O so that the quantified formula is decidable by polyPH^O.","section":"Lemma 3.2"},{"comment":"The theorem statement says 'with high probability' but does not quantify the probability or the exact measure over S(w*); the proof later uses 2/3, so this should be stated as a specific parameter in the theorem.","section":"Theorem 4.3"},{"comment":"The diagonalization should specify that n_i is chosen sufficiently large so that no previously handled verifier can query U_{n_i} on its own input length; otherwise fixing U_{n_i} could disturb earlier diagonalization steps. This is fixable by a standard size argument but should be stated.","section":"Theorem 4.4"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses an interesting and timely question, and the core idea of using the AK07 oracle to separate a bounded-error quantum-classical hierarchy is appealing. The definitional problem with polyQCPH and the unsupported Eq. (1) are the main obstacles. The stress-test concern about the moment mismatch in Eq. (1) is partly mitigable because the squared state distance is bounded by a constant, so a first-moment bound would imply the printed second-moment bound up to constants; the real gap is the lack of a proof of any first-moment bound against adaptive queries. If the authors can supply a complete derivation or a precise citation to the relevant AK07 theorem, and fix the other issues, the paper could become a solid contribution. I would not recommend rejection, as the central claims are likely correct and the errors appear fixable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper is worth sending to a serious referee, but not for publication as is. The main new thing is the class polyQCPH, a bounded-error quantum-classical hierarchy with polynomially many alternating classical proofs. The authors show QMA sits inside polyQCPH for classical oracles (because it equals PSPACE there) but not for a specific quantum unitary oracle, and they claim the same for a distributional oracle. If correct, that resolves the bounded-error version of an open problem from Aaronson 2009. The classical containment is straightforward, and the quantum oracle is the AK07 one, so the overall architecture is plausible.\n\nWhat the paper does well: it gives a clear motivation for why polyQCPH is the right bounded-error surrogate for PSPACE in this setting, and Section 4.3 has a useful discussion of why the classical oracle separations for QMA vs QCMA don't automatically extend. That kind of context matters.\n\nThe soft spots are in the proofs. In Theorem 4.3, the key inequality (1) asserts a bound on the squared Euclidean distance between final states using a first-moment-looking right-hand side, and the derivation from the AK07 geometric lemma is not given. The stress-test is right that as printed, the Markov step in (2) doesn't follow. This is load-bearing because the diagonalization in Theorem 4.4 rests entirely on it. A careful referee would need the correct hybrid argument and the proper concentration step. My guess is this is salvageable from AK07, but it must be re-derived.\n\nLemma 4.1 also has a plain error: running the QMA verifier on the maximally mixed proof gives acceptance probability at most about 2^{-p(n)} on YES instances, not the >1/2 gap PQP needs. The statement QMA ⊆ PQP is true, but the proof given here is not it. The distributional proof in Theorem 5.3 has a similar problem: Lemma 5.2 bounds the distinguishing bias for each fixed witness, but the event E1 involves an existential quantifier, and the paper's Markov step doesn't account for the exponential number of witnesses. Again, probably fixable, but not as written.\n\nCitations look appropriate; the paper leans on AK07, LLPY24, MW05, and standard PSPACE facts. No sign of fitting or invented parameters; polyQCPH is a new class but defined honestly.\n\nWho is this for: people who care about oracle separations and the QMA vs QCMA landscape. It's a warning about reading too much into non-standard oracle models. I'd bring it to a reading group, but I'd tell them ahead of time that the proof gaps are the discussion topic.\n\nRecommendation: send to a serious referee, but require revision. The referee should ask for a corrected Eq. (1), a real proof of QMA^U ⊆ PQP^U, and a cleaner argument in Theorem 5.3. The core claim is likely true, and the paper deserves the time.","headline":"Introduces polyQCPH and claims a bounded-error quantum oracle separation that would resolve an Aaronson open problem; the idea is good, but the write-up has fixable proof gaps.","tokens_in":19757,"tokens_out":5666,"would_cite":false,"duration_ms":55976,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68Q15","68Q12","81P68"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a bounded-error containment, QMA ⊆ polyQCPH, that holds for every classical oracle but fails for a specific quantum unitary oracle.","keywords":["QMA","polyQCPH","quantum-classical polynomial hierarchy","quantum oracle","distributional oracle","oracle separation","bounded-error complexity","relativization"],"falsifier":"Compute the expectation in Theorem 4.3, Eq. (1), for a small query algorithm (for example, a two-query circuit) on the [AK07] oracle with $n = 20$ qubits and witness length $m = 100$: the paper bounds the expected squared distance $E\\bigl[\\bigl\\||\\Phi_\\psi\\rangle - |\\Phi_I\\rangle\\bigr\\|^2\\bigr]$ by $O\\bigl(T\\sqrt{(m+1)/2^n}\\bigr)$, a quantity with the scaling of a distance. If the hybrid method actually accumulates $T^2(m+1)/2^n$ (squared-distance scaling), then the Markov step in Eq. (2) does not give $o(1)$ probability for polynomial $T$, and the diagonalization cannot force error. Evaluating the expectation exactly for one- and two-query algorithms would settle the exponent and hence whether $\\mathrm{QMA}^U \\not\\subseteq \\mathrm{polyQCPH}^U$ is established.","tokens_in":45,"feed_emoji":"⚛️","tokens_out":15634,"duration_ms":194098,"temperature":0.7,"pith_summary":"The paper establishes that the containment $\\mathrm{QMA} \\subseteq \\mathrm{polyQCPH}$—quantum Merlin–Arthur inside the polynomial-time quantum-classical hierarchy with polynomially many alternating classical proofs—holds in the unrelativized model and relative to every classical oracle, yet there is a single quantum unitary oracle $U$ relative to which $\\mathrm{QMA}^U$ is not contained in $\\mathrm{polyQCPH}^U$. This is the first bounded-error example of a complexity-class containment that relativizes with respect to all classical oracles but fails with respect to a quantum oracle, resolving the open problem from [Aar09]. The proof works by defining $\\mathrm{polyQCPH}$ as a bounded-error analogue of $\\mathrm{PSPACE}$ in the quantum-oracle setting, then showing that the oracle of [AK07], already known to separate $\\mathrm{QMA}$ from $\\mathrm{QCMA}$, also separates $\\mathrm{QMA}$ from $\\mathrm{polyQCPH}$. The same separation is shown to hold relative to a distributional oracle, which the authors offer as a caution about the use of such non-standard oracle models.","feed_headline":"Quantum oracle breaks a containment all classical oracles keep","feed_subtitle":"QMA sits inside polyQCPH for every classical oracle, but one quantum oracle separates them—the first bounded-error case.","key_machinery":"The central object is $\\mathrm{polyQCPH}$, the quantum-classical polynomial hierarchy with a polynomial number of alternating classical proofs, defined so that it equals $\\mathrm{PSPACE}$ relative to classical oracles but remains a bounded-error quantum class with a meaningful quantum-oracle relativization. The load-bearing mechanism is the geometric lemma for $p$-uniform measures: for a measure $\\sigma$ obtained by conditioning Haar measure on an event of probability at least $p$, $E_{|\\psi\\rangle\\sim\\sigma}[\\langle\\psi|\\rho|\\psi\\rangle] = O((1+\\log(1/p))/N)$ for any density matrix $\\rho$ on $N$ dimensions. Applied through the hybrid method, this bounds how well a $T$-query algorithm can distinguish the oracle $U_\\psi$ from the identity when the good witness $w^*$ has a cell $S(w^*)$ of measure at least $2^{-m}$; the resulting small per-witness bias is what permits the diagonalization in Theorem 4.4 to make every $\\mathrm{polyQCPH}$ verifier err on some input. The same bound extends to verifiers with query access to $U^*$, $U^T$, and $U^{-1}$.","core_discovery":"The paper introduces the bounded-error quantum class $\\mathrm{polyQCPH}$, in which a polynomial-time quantum verifier receives polynomially many alternating classical proofs and must accept true instances with probability at least $2/3$ and reject false instances with probability at most $1/3$. It proves that $\\mathrm{QMA} \\subseteq \\mathrm{polyQCPH}$, that $\\mathrm{QMA}^O \\subseteq \\mathrm{polyQCPH}^O$ for every classical oracle $O$—because $\\mathrm{polyQCPH}^O = \\mathrm{PSPACE}^O$—and that there exists a quantum unitary oracle $U$ such that $\\mathrm{QMA}^U \\not\\subseteq \\mathrm{polyQCPH}^U$. The separating $U$ is the reflection oracle $U_\\psi = I - 2|\\psi\\rangle\\langle\\psi|$ for a Haar-random state $|\\psi\\rangle$ versus the identity $I$, the same oracle used in [AK07] to separate $\\mathrm{QMA}$ from $\\mathrm{QCMA}$. The argument adapts the geometric bound on $p$-uniform measures from [AK07] to show that a witness $w^*$ whose cell $S(w^*)$ has large measure forces every classical witness to have small distinguishing bias between $U_\\psi$ and $I$; a diagonalization over $\\mathrm{polyQCPH}$ verifiers then constructs $U$ with a verifier that errs on a chosen input. The same construction is carried out for the distributional oracle model using the oracle of [LLPY24], yielding $\\mathrm{QMA}^D \\not\\subseteq \\mathrm{polyQCPH}^D$ for a distributional oracle $D$.","pith_inferences":["A direct corollary the paper leaves implicit: since $\\mathrm{QMA}^U \\subseteq \\mathrm{PQP}^U$ and $\\mathrm{PQP}^U \\not\\subseteq \\mathrm{polyQCPH}^U$, the $\\mathrm{PQP}$/$\\mathrm{PSPACE}$ analogue also fails to relativize quantumly, so the phenomenon is not an artifact of quantum proofs.","The geometric-bound mechanism suggests a recipe: any quantum oracle problem with a single NO oracle and a good-witness partition into cells of measure at least $2^{-m}$ should yield a $\\mathrm{QMA}$-vs-$\\mathrm{polyQCPH}$ separation; testing this on other single-NO oracles from quantum cryptography could show whether the partition structure is the key driver.","If the geometric bound's exponent were corrected to a different $T$-dependence, the separation might shift from $\\mathrm{polyQCPH}$ to a class with fewer alternation rounds; a natural stress test is whether the [AK07] oracle still separates $\\mathrm{QMA}$ from $\\mathrm{QCPH}$ (constant alternations).","The distributional separation is driven by the prover's ignorance of the sampled $r_n$: the noise-robust witness bound of [LLPY24] (Lemma 5.2) holds only when proofs avoid $r_n$. One could test whether allowing classical proofs to depend on $r_n$ either restores the containment or requires a different oracle construction."],"forward_implications":["It resolves the open problem from [Aar09] for bounded-error classes: $\\mathrm{QMA} \\subseteq \\mathrm{polyQCPH}$ is a containment that relativizes for every classical oracle but not for a quantum unitary oracle, so relativizing proof techniques cannot settle it in both settings at once.","The [AK07] reflection oracle separates $\\mathrm{QMA}$ not only from $\\mathrm{QCMA}$, but from the entire polynomial quantum-classical hierarchy, and from $\\mathrm{PQP}$ as well, since $\\mathrm{QMA}^U \\subseteq \\mathrm{PQP}^U$ (Lemma 4.1).","The separation persists even when the $\\mathrm{polyQCPH}$ verifier is allowed queries to $U^*$, $U^T$, and $U^{-1}$, so it does not rely on restricted query access.","For distributional oracles, the equality $\\mathrm{polyQCPH}^D = \\mathrm{PSPACE}^D$ that holds for classical oracles breaks: there is a distributional oracle $D$ with $\\mathrm{QMA}^D \\not\\subseteq \\mathrm{polyQCPH}^D$.","The results support the abstract's caution: quantum unitary oracles and distributional oracles must be used carefully, because separations in these models do not automatically translate into relativization barriers for classical proof techniques."],"supporting_citations":[{"why":"Supplies the quantum oracle problem ($U_\\psi$ versus $I$) and the geometric lemma (Lemma 2.13) that Theorem 4.3 adapts to bound the distinguishing bias.","marker":"[AK07]"},{"why":"Poses the open problem for bounded-error classes and provides the zero-error precedent (ZQEXP) that this paper extends.","marker":"[Aar09]"},{"why":"Supplies the distributional oracle problem and Lemma 5.2, the distinguishing-bias bound for classical witnesses used in the distributional separation.","marker":"[LLPY24]"},{"why":"Provides $\\mathrm{QMA} \\subseteq \\mathrm{PSPACE}$ and the strong error reduction used to show $\\mathrm{QMA}^U \\subseteq \\mathrm{PQP}^U$ (Lemma 4.1).","marker":"[MW05]"},{"why":"Defines QCPH, the quantum-classical polynomial hierarchy that polyQCPH generalizes.","marker":"[GSSSY22]"},{"why":"Introduces the distributional oracle model, the framework for the paper's distributional separation.","marker":"[NN24]"},{"why":"Supplies the TQBF-completeness of PSPACE (Lemma 3.1) used to prove $\\mathrm{polyQCPH}^O = \\mathrm{PSPACE}^O$ for classical oracles.","marker":"[AB09]"}],"fun_headline_variants":["Quantum oracle splits QMA from polyQCPH","Bounded-error oracle separation: QMA not in polyQCPH","QMA escapes polyQCPH via quantum oracle","Quantum oracle breaks classical containment","First bounded-error quantum oracle separation"],"cache_read_input_tokens":21760,"weakest_assumption_plain":"The separation stands or falls on the geometric bound from [AK07]: a witness cell of large $p$-uniform measure must force every classical proof to have small distinguishing bias between $U_\\psi$ and the identity, in the exact quantitative form used in Theorem 4.3; if that bound, or its stated exponent, fails, the diagonalization in Theorem 4.4 cannot force a $\\mathrm{polyQCPH}$ verifier to err on the selected state.","fun_headline_variants_meta":{"raw":{"variants":["Quantum oracle splits QMA from polyQCPH","Bounded-error oracle separation: QMA not in polyQCPH","QMA escapes polyQCPH via quantum oracle","Quantum oracle breaks classical containment","First bounded-error quantum oracle separation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000426,"raw_usage":{"total_tokens":2288,"prompt_tokens":1158,"completion_tokens":1130,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":774,"completion_tokens_details":{"reasoning_tokens":1059}},"tokens_in":774,"tokens_out":1130,"duration_ms":9365,"temperature":1.0,"reasoning_tokens":1059,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:55:39.420624+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the expectation in Theorem 4.3, Eq. (1), for a small query algorithm (for example, a two-query circuit) on the [AK07] oracle with $n = 20$ qubits and witness length $m = 100$: the paper bounds the expected squared distance $E\\bigl[\\bigl\\||\\Phi_\\psi\\rangle - |\\Phi_I\\rangle\\bigr\\|^2\\bigr]$ by $O\\bigl(T\\sqrt{(m+1)/2^n}\\bigr)$, a quantity with the scaling of a distance. If the hybrid method actually accumulates $T^2(m+1)/2^n$ (squared-distance scaling), then the Markov step in Eq. (2) does not give $o(1)$ probability for polynomial $T$, and the diagonalization cannot force error. Evaluating the expectation exactly for one- and two-query algorithms would settle the exponent and hence whether $\\mathrm{QMA}^U \\not\\subseteq \\mathrm{polyQCPH}^U$ is established.","supporting_citations":[],"review_version":1}