Global Bounds beyond Local Quantum Metrology

Quantum Cram\'er--Rao theory is intrinsically local: it bounds precision near a specified parameter value, and its saturating measurement generally depends on that value. Barankin-type bounds use finite parameter displacements, but remain anchored to a chosen reference value. This leaves open a basic global-estimation problem: when the parameter is known only within a broad domain, what precision can be guaranteed by a single estimator and a single measurement strategy fixed before the true value is localized? We answer this question by introducing global score functions tied to a weighted variance over the whole parameter domain. Their correlations generate a hierarchy of precision bounds: global Cram\'er--Rao and Barankin-type bounds arise as restricted levels, whereas unrestricted score correlations yield a fully global bound for the prescribed weighted variance. The hierarchy recovers local Cram\'er--Rao theory in the many-repetition limit and reveals genuinely global precision limits for finite data over broad domains. In the quantum setting, the construction identifies when this fully global bound can be realized by a single parameter-independent measurement. The same framework extends to Bayesian estimation, recovering the Van Trees bound in the local limit while yielding stronger finite-width lower bounds on the Bayesian mean-square error beyond this limit.