  Motivated by quantum resource theories, we introduce a notion of
incompatibility for quantum measurements relative to a reference basis. The
notion arises by considering states diagonal in that basis and investigating
whether probability distributions associated with different quantum
measurements can be converted into one another by probabilistic
post-processing. The induced preorder over quantum measurements is directly
related to multivariate majorization and gives rise to families of monotones,
i.e., scalar quantifiers that preserve the ordering. For the case of orthogonal
measurement we establish a quantitative connection between incompatibility,
quantum coherence and entropic uncertainty relations. We generalize the
construction to arbitrary POVM measurements and report complete families of
monotones.
