The \emph{sum-of-squares (SoS) complexity} of a $d$-multiquadratic polynomial $f$ (quadratic in each of $d$ blocks of $n$ variables) is the minimum $s$ such that $f = \sum_{i=1}^s g_i^2$ with each $g_i$ $d$-multilinear. In the case $d=2$, Hrubeš, Wigderson and Yehudayoff (2011) showed that an $n^{1+Ω(1)}$ lower bound on the SoS complexity of explicit biquadratic polynomials implies an exponential lower bound for non-commutative arithmetic circuits. In this paper, we establish an analogous connection between general \emph{multiquadratic sum-of-squares} and \emph{commutative arithmetic formulas}. Specifically, we show that an $n^{d-o(\log d)}$ lower bound on the SoS complexity of explicit $d$-multiquadratic polynomials, for any $d = d(n)$ with $ω(1) \le d(n) \le O(\frac{\log n}{\log\log n})$, would separate the algebraic complexity classes VNC$^1$ and VNP.

翻译：一个$d$重二次多项式$f$（在$d$组每组$n$个变量的块中均为二次型）的\\emph{平方和（SoS）复杂度}是指满足$f = \\sum_{i=1}^s g_i^2$的最小值$s$，其中每个$g_i$为$d$重线性多项式。在$d=2$的情形下，Hrubeš、Wigderson与Yehudayoff（2011）证明了对显式双二次多项式SoS复杂度的$n^{1+Ω(1)}$下界将蕴含非交换算术电路存在指数级下界。本文建立了\\emph{多重二次平方和}与\\emph{交换算术公式}之间的类比关联。具体而言，我们证明了对任意满足$ω(1) \\le d(n) \\le O(\\frac{\\log n}{\\log\\log n})$的$d = d(n)$，若显式$d$重二次多项式的SoS复杂度存在$n^{d-o(\\log d)}$下界，则将分离代数复杂度类VNC$^1$与VNP。