  The existence of incompatible measurements, epitomized by Heisenberg's
uncertainty principle, is one of the distinctive features of quantum theory. So
far, quantum incompatibility has been studied for measurements that test the
preparation of physical systems. Here we extend the notion to measurements that
test dynamical processes, possibly consisting of multiple time steps. Such
measurements are known as testers and are implemented by interacting with the
tested process through a sequence of state preparations, interactions, and
measurements. Our first result is a characterization of the incompatibility of
quantum testers, for which we provide necessary and sufficient conditions.
Then, we propose a quantitative measure of incompatibility. We call this
measure the robustness of incompatibility and define it as the minimum amount
of noise that has to be added to a set of testers in order to make them
compatible. We show that (i) the robustness is lower bounded by the
distinguishability of the sequence of interactions used by the tester and (ii)
maximum robustness is attained when the interactions are perfectly
distinguishable. The general results are illustrated in the concrete example of
binary testers probing the time-evolution of a single-photon polarization.
