Quantum theory offers measurement incompatibility, that is, the existence of
quantum measurements that cannot be carried out simultaneously on single
systems. Measurement incompatibility is essential for probing many aspects of
quantum correlations and quantum information processing. However, its
fundamental and generic link with nonclassical correlations observed in the
simplest prepare-and-measure scenario is still untold. In the
prepare-and-measure scenario, we uncover that d-dimensional classical systems
assisted with shared randomness reproduce all the input-output statistics
obtained from any set of d-dimensional compatible quantum measurements. Thus,
any quantum advantage in one-way communication tasks with d-dimensional systems
witnesses incompatibility of the measurements on the receiver's end in a
semi-device-independent way. To witness incompatibility of an arbitrary number
of quantum measurements acting on an arbitrary dimension, wherein different
measurements have different outcomes, we introduce a class of communication
tasks - a general version of random access codes. We provide generic upper
bounds on the success metric of these tasks for compatible measurements. These
bounds are tight whenever the dimension on which the measurements act is not
larger than the number of outcomes of any of the measurements