Strict hierarchy between $n$-wise measurement simulability, compatibility structures, and multi-copy compatibility

The incompatibility of quantum measurements, i.e. the fact that certain observable quantities cannot be measured jointly is widely regarded as a distinctive quantum feature with important implications for the foundations and the applications of quantum information theory. While the standard incompatibility of multiple measurements has been the focus of attention since the inception of quantum theory, its generalizations, such as measurement simulability, $n$-wise incompatibility, and mulit-copy incompatibility have only been proposed recently. Here, we point out that all these generalizations are differing notions of the question of how many measurements are genuinely contained in a measurement device. We then show, that all notions do differ not only in their operational meaning but also mathematically in the set of measurement assemblages they describe. We then fully resolve the relations between these different generalizations, by showing a strict hierarchy between these notions. Hence, we provide a general framework for generalized measurement incompatibility. Finally, we consider the implications our results have for recent works using these different notions.