  Hypercontractive inequalities have become important tools in theoretical
computer science and have recently found applications in quantum computation.
In this note we discuss how hypercontractive inequalities, in various settings,
can be used to obtain (fairly) concise proofs of several results in quantum
information theory: a recent lower bound of Lancien and Winter on the bias
achievable by local measurements which are 4-designs; spectral concentration
bounds for k-local Hamiltonians; and a recent result of Pellegrino and
Seoane-Sepulveda giving general lower bounds on the classical bias obtainable
in multiplayer XOR games.
