  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 -- 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 doesn't trivialise 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.
