  For any $K>2$ and the multiplicative cyclic group $\Omega_K$ of order $K$,
consider any function $f:\Omega_K^n\to\mathbf{C}$ and its Fourier expansion
$f(z)=\sum_{\alpha\in\{0,1,\ldots,K-1\}^n}a_\alpha z^\alpha$, with
$d:=\text{deg}(f)$ denoting its degree as a multivariate polynomial.
  We prove a Bohnenblust--Hille (BH) inequality in this setting: the
$\ell_{2d/(d+1)}$ norm of the Fourier coefficients of $f$ is bounded by
$C(d,K)\|f\|_\infty$ with $C(d,K)$ independent of $n$.
  This is the interpolating case between the now well-understood BH
inequalities for functions on the poly-torus ($K =\infty$) and the hypercube
($K=2$) but those extreme cases of $K$ have special properties whose absence
for intermediate $K$ prevent a proof by the standard BH framework.
  New techniques are developed exploiting the group structure of $\Omega_K^n$.
  By known reductions, the cyclic group BH inequality also entails a
noncommutative BH inequality for tensor products of the $K \times K$ complex
matrix algebra (or in the language of quantum mechanics, systems of $K$-level
qudits). These new BH inequalities generalize several applications in harmonic
analysis and statistical learning theory to broader classes of functions and
operators.
