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We derive lower bounds on the variance of estimators in quantum metrology bychoosing test observables that define constraints on the unbiasedness of theestimator. The quantum bounds are obtained by analytical optimization over allpossible quantum measurements and estimators that satisfy the givenconstraints. We obtain hierarchies of increasingly tight bounds that includethe quantum Cram\'er-Rao bound at the lowest order. In the opposite limit, thequantum Barankin bound is the variance of the locally best unbiased estimatorin quantum metrology. Our results reveal generalizations of the quantum Fisherinformation that are able to avoid regularity conditions and identify thresholdbehavior in quantum measurements with mixed states, caused by finite data.

Hierarchies of Frequentist Bounds for Quantum Metrology: From Cram\'er-Rao to Barankin