  We introduce the notion of incompatibility witness for quantum channels,
defined as an affine functional that is non-negative on all pairs of compatible
channels and strictly negative on some incompatible pair. This notion extends
the recent definition of incompatibility witnesses for quantum measurements. We
utilize the general framework of channels acting on arbitrary finite
dimensional von Neumann algebras, thus allowing us to investigate
incompatibility witnesses on measurement-measurement, measurement-channel and
channel-channel pairs. We prove that any incompatibility witness can be
implemented as a state discrimination task in which some intermediate classical
information is obtained before completing the task. This implies that any
incompatible pair of channels gives an advantage over compatible pairs in some
such state discrimination task.
