  I propose quantum versions of the Ziv-Zakai bounds as alternatives to the
widely used quantum Cram\'er-Rao bounds for quantum parameter estimation. From
a simple form of the proposed bounds, I derive both a "Heisenberg" error limit
that scales with the average energy and a limit similar to the quantum
Cram\'er-Rao bound that scales with the energy variance. These results are
further illustrated by applying the bound to a few examples of optical phase
estimation, which show that a quantum Ziv-Zakai bound can be much higher and
thus tighter than a quantum Cram\'er-Rao bound for states with highly
non-Gaussian photon-number statistics in certain regimes and also stay close to
the latter where the latter is expected to be tight.
