  A missing piece in quantum information theory, with very few exceptions, has
been to provide the random coding exponents for quantum information-processing
protocols. We remedy the situation by providing these exponents for a variety
of protocols including those at the top of the family tree of protocols. Our
line of attack is to provide an exponential bound on the decoupling error for a
restricted class of completely positive maps where a key term in the exponent
is in terms of a R\'enyi \alpha-information-theoretic quantity for any \alpha
$\in$ (1,2]. Among the protocols covered are fully quantum Slepian-Wolf,
quantum state merging, quantum state redistribution, quantum/classical
communication across channels with side information at the transmitter with or
without entanglement assistance, and quantum communication across broadcast
channels.
