  We extend the recent bounds of Sason and Verd\'u relating R\'enyi entropy and
Bayesian hypothesis testing [arXiv:1701.01974] to the quantum domain and show
that they have a number of different applications. First, we obtain a sharper
bound relating the optimal probability of correctly distinguishing elements of
an ensemble of states to that of the pretty good measurement, and an analogous
bound for optimal and pretty good entanglement recovery. Second, we obtain
bounds relating optimal guessing and entanglement recovery to the fidelity of
the state with a product state, which then leads to tight tripartite
uncertainty and monogamy relations.
