  We prove a moment majorization principle for matrix-valued functions with
domain $\{-1,1\}^{m}$, $m\in\mathbb{N}$. The principle is an inequality between
higher-order moments of a non-commutative multilinear polynomial with different
random matrix ensemble inputs, where each variable has small influence and the
variables are instantiated independently.
  This technical result can be interpreted as a noncommutative generalization
of one of the two inequalities of the seminal invariance principle of Mossel,
O'Donnell and Oleszkiewicz. Applications to noncommutative noise stability and
noncommutative anticoncentration are given.
