  A hierarchy of incompatibility measures, each corresponding to a set of
allowed measurements, is introduced. These quantify the genuine quantum
incompatibility in the estimation task of multiple parameters, and have a
geometric character which is backed by a clear operational interpretation. The
analysis is then applied to some simple systems of one, two, and three qubits
in order to track the effect of a local depolarizing noise on the
incompatibility of the estimation task. A semidefinite program is described and
used to numerically compute the figure of merit when the analytical tools are
not sufficient, among these we include an upper bound computable from the
symmetric logarithmic derivatives only. Finally we generalize the results
obtained for one and two qubits to the case of a general unitary encoding on a
finite dimensional probe.
