  We prove a hypercontractive inequality for matrix-valued functions defined
over large alphabets, generalizing the result of Ben-Aroya, Regev, de Wolf
(FOCS'08) for the Boolean alphabet. For such we prove a generalization of the
$2$-uniform convexity inequality of Ball, Carlen, Lieb (Inventiones
Mathematicae'94). Using our inequality, we present upper and lower bounds for
the communication complexity of Hidden Hypermatching when defined over large
alphabets, which generalizes the well-known Boolean Hidden Matching problem. We
then consider streaming algorithms for approximating the value of Unique Games
on a $t$-hyperedge hypergraph: an edge-counting argument gives an
$r$-approximation with $O(\log{n})$ space. On the other hand, via our
communication lower bound we show that every streaming algorithm in the
adversarial model achieving a $(r-\varepsilon)$-approximation requires
$\Omega(n^{1-2/t})$ quantum space. This generalizes the seminal work of
Kapralov, Khanna, Sudan (SODA'15), and expand to the quantum setting results
from Kapralov, Krachun (STOC'19) and Chou et al. (STOC'22).
  We next present a lower bound for locally decodable codes ($\mathsf{LDC}$)
over large alphabets. An $\mathsf{LDC}$ $C:\mathbb{Z}_r^n\to \mathbb{Z}_r^N$ is
an encoding of $x$ into a codeword in such a way that one can recover an
arbitrary $x_i$ (with probability at least $1/r+\varepsilon$) by making a few
queries to a corrupted codeword. The main question here is the trade-off
between $N$ and $n$. Via hypercontractivity, we give an exponential lower bound
$N= 2^{\Omega(\varepsilon^4 n/r^4)}$ for $2$-query (possibly non-linear)
$\mathsf{LDC}$s over $\mathbb{Z}_r$ and using the non-commutative Khintchine
inequality we improved our bound to $N= 2^{\Omega(\varepsilon^2 n/r^2)}$.
Previously exponential lower bounds were known for $r=2$ (Kerenidis, de Wolf
(JCSS'04)) and linear codes (Dvir, Shpilka (SICOMP'07)).
