  One of the basic distinctions between classical and quantum mechanics is the
existence of fundamentally incompatible quantities. Such quantities are present
on all levels of quantum objects: states, measurements, quantum channels, and
even higher order dynamics. In this manuscript, we show that two seemingly
different aspects of quantum incompatibility: the quantum marginal problem of
states and the incompatibility on the level of quantum channels are in
many-to-one correspondence. Importantly, as incompatibility of measurements is
a special case of the latter, it also forms an instance of the quantum marginal
problem. The generality of the connection is harnessed by solving the marginal
problem for Gaussian and Bell diagonal states, as well as for pure states under
depolarizing noise. Furthermore, we derive entropic criteria for channel
compatibility, and develop a converging hierarchy of semi-definite programs for
quantifying the strength of quantum memories.
