  Measurement incompatibility stipulates the existence of quantum measurements
that cannot be carried out simultaneously on single systems. We show that the
set of input-output probabilities obtained from d-dimensional classical systems
assisted with shared randomness is the same as the set obtained from
d-dimensional quantum strategies restricted to compatible measurements with
shared randomness in any communication scenario. Thus, measurement
incompatibility is necessary for quantum advantage in communication, and any
quantum advantage (with or without shared randomness) in communication acts as
a witness to the incompatibility of the measurements at the receiver's end in a
semi-device-independent way. We introduce a class of communication tasks - a
general version of random access codes - to witness incompatibility of an
arbitrary number of quantum measurements with arbitrary outcomes acting on
d-dimensional systems, and provide generic upper bounds on the success metric
of these tasks for compatible measurements. We identify all sets of three
incompatible rank-one projective qubit measurements that random access codes
can witness. Finally, we present the generic relationship between different
sets of probability distributions - classical, quantum with or without shared
randomness, and quantum restricted to compatible measurements with or without
shared randomness - produced in communication scenarios.
