A CSP whose quantum endomorphism monoid is non-classical admits no commutativity gadget; in particular, k-colouring for k at least 4 has no commutativity gadget, while an oracular commutativity gadget exists.
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Existence and nonexistence of commutativity gadgets for entangled CSPs
A CSP whose quantum endomorphism monoid is non-classical admits no commutativity gadget; in particular, k-colouring for k at least 4 has no commutativity gadget, while an oracular commutativity gadget exists.