  We extend quantum Stein's lemma in asymmetric quantum hypothesis testing to
composite null and alternative hypotheses. As our main result, we show that the
asymptotic error exponent for testing convex combinations of quantum states
$\rho^{\otimes n}$ against convex combinations of quantum states
$\sigma^{\otimes n}$ can be written as a regularized quantum relative entropy
formula. We prove that in general such a regularization is needed but also
discuss various settings where our formula as well as extensions thereof become
single-letter. This includes an operational interpretation of the relative
entropy of coherence in terms of hypothesis testing. For our proof, we start
from the composite Stein's lemma for classical probability distributions and
lift the result to the non-commutative setting by using elementary properties
of quantum entropy. Finally, our findings also imply an improved recoverability
lower bound on the conditional quantum mutual information in terms of the
regularized quantum relative entropy -- featuring an explicit and universal
recovery map.
