  Quantum measurements based on mutually unbiased bases are commonly used in
quantum information processing, as they are generally viewed as being maximally
incompatible and complementary. Here we quantify precisely the degree of
incompatibility of mutually unbiased bases (MUB) using the notion of noise
robustness. Specifically, for sets of $k$ MUB in dimension $d$, we provide
upper and lower bounds on this quantity. Notably, we get a tight bound in
several cases, in particular for complete sets of $k=d+1$ MUB (given $d$ is a
prime power). On the way, we also derive a general upper bound on the noise
robustness for an arbitrary set of quantum measurements. Moreover, we prove the
existence of sets of $k$ MUB that are operationally inequivalent, as they
feature different noise robustness, and we provide a lower bound on the number
of such inequivalent sets up to dimension 32. Finally, we discuss applications
of our results for Einstein-Podolsky-Rosen steering.
