Complexity of compatible measurements

Measurement incompatibility is one of the basic aspects of quantum theory. Here we study the structure of the set of compatible, i.e., jointly measurable, measurements. We are interested in whether or not there exist compatible measurements whose parent is maximally complex, in the sense of requiring a number of outcomes exponential in the number of measurements, and related questions. Although we show this to be the case in a number of simple scenarios, we show that generically it cannot happen, by proving an upper bound on the number of outcomes of a parent measurement that is linear in the number of compatible measurements. We discuss why this does not trivialize the problem of finding parent measurements, but rather shows that a trade-off between memory and time can be achieved. Finally, we also investigate the complexity of extremal compatible measurements in regimes where our bound is not tight and uncover rich structure.