  The advantage that quantum systems provide for certain quantum information
processing tasks over their classical counterparts can be quantified within the
general framework of resource theories. Certain distance functions between
quantum states have successfully been used to quantify resources like
entanglement and coherence. Perhaps surprisingly, such a distance-based
approach has not been adopted to study resources of quantum measurements, where
other geometric quantifiers are used instead. Here, we define distance
functions between sets of quantum measurements and show that they naturally
induce resource monotones for convex resource theories of measurements. By
focusing on a distance based on the diamond norm, we establish a hierarchy of
measurement resources and derive analytical bounds on the incompatibility of
any set of measurements. We show that these bounds are tight for certain
projective measurements based on mutually unbiased bases and identify scenarios
where different measurement resources attain the same value when quantified by
our resource monotone. Our results provide a general framework to compare
distance-based resources for sets of measurements and allow us to obtain
limitations on Bell-type experiments.
