  The laws of quantum mechanics place fundamental limits on the accuracy of
measurements and therefore on the estimation of unknown parameters of a quantum
system. In this work, we prove lower bounds on the size of confidence regions
reported by any region estimator for a given ensemble of probe states and
probability of success. Our bounds are derived from a previously unnoticed
connection between the size of confidence regions and the error probabilities
of a corresponding binary hypothesis test. In group-covariant scenarios, we
find that there is an ultimate bound for any estimation scheme which depends
only on the representation-theoretic data of the probe system, and we evaluate
its asymptotics in the limit of many systems, establishing a general
"Heisenberg limit" for region estimation. We apply our results to several
examples, in particular to phase estimation, where our bounds allow us to
recover the well-known Heisenberg and shot-noise scaling.
