  We study the problem of binary composite channel discrimination in the
asymmetric setting, where the hypotheses are given by fairly arbitrary sets of
channels, and samples do not have to be identically distributed. In the case of
quantum channels we prove: (i) a characterization of the Stein exponent for
parallel channel discrimination strategies and (ii) an upper bound on the Stein
exponent for adaptive channel discrimination strategies. We further show that
already for classical channels this upper bound can sometimes be achieved and
be strictly larger than what is possible with parallel strategies. Hence, there
can be an advantage of adaptive channel discrimination strategies with
composite hypotheses for classical channels, unlike in the case of simple
hypotheses. Moreover, we show that classically this advantage can only exist if
the sets of channels corresponding to the hypotheses are non-convex. As a
consequence of our more general treatment, which is not limited to the
composite i.i.d. setting, we also obtain a generalization of previous composite
state discrimination results.
