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)).

翻译：我们证明了大字母表上矩阵值函数的超压缩不等式，推广了Ben-Aroya、Regev和de Wolf（FOCS'08）针对布尔字母表的结果。为此，我们证明了Ball、Carlen、Lieb（Inventiones Mathematicae'94）的$2$-一致凸性不等式的推广形式。利用我们的不等式，我们给出了在大字母表上定义的隐藏超匹配问题的通信复杂性的上下界，该问题推广了著名的布尔隐藏匹配问题。接着，我们考虑用于近似$t$-超边超图上唯一博弈值的流式算法：通过边计数论证，我们得到一种$r$-近似算法，空间复杂度为$O(\log{n})$。另一方面，通过我们的通信下界，我们证明在对抗模型下，任何达到$(r-\varepsilon)$-近似的流式算法都需要$\Omega(n^{1-2/t})$的量子空间。这一结果推广了Kapralov、Khanna和Sudan（SODA'15）的开创性工作，并将Kapralov、Krachun（STOC'19）以及Chou等人（STOC'22）的结果扩展到量子场景。接下来，我们给出了大字母表上局部可译码（$\mathsf{LDC}$）的下界。一个$\mathsf{LDC}$ $C:\mathbb{Z}_r^n\to \mathbb{Z}_r^N$是将$x$编码为码字的映射，使得通过少量查询被损坏的码字即可恢复任意$x_i$（概率至少为$1/r+\varepsilon$）。核心问题在于$N$与$n$之间的权衡。通过超压缩性，我们给出了$\mathbb{Z}_r$上2-查询（可能非线性）$\mathsf{LDC}$的指数下界$N= 2^{\Omega(\varepsilon^4 n/r^4)}$，并利用非交换Khintchine不等式将下界改进为$N= 2^{\Omega(\varepsilon^2 n/r^2)}$。此前，仅对于$r=2$（Kerenidis、de Wolf（JCSS'04））和线性码（Dvir、Shpilka（SICOMP'07））已知指数下界。