  We provide moment bounds for expressions of the type $(X^{(1)} \otimes \dots
\otimes X^{(d)})^T A (X^{(1)} \otimes \dots \otimes X^{(d)})$ where $\otimes$
denotes the Kronecker product and $X^{(1)}, \dots, X^{(d)}$ are random vectors
with independent, mean 0, variance 1, subgaussian entries. The bounds are tight
up to constants depending on $d$ for the case of Gaussian random vectors. Our
proof also provides a decoupling inequality for expressions of this type. Using
these bounds, we obtain new, improved concentration inequalities for
expressions of the form $\|B (X^{(1)} \otimes \dots \otimes X^{(d)})\|_2$.
