Multiquadratic Sum-of-Squares Lower Bounds Imply VNC${}^1$ $\ne$ VNP

Abstract

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 $n^{1+\Omega (1)}$ 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 $\omega (1)\le d(n)\le O(\frac{\log n}{\log \log n})$, would separate the algebraic complexity classes VNC${}^1$ and VNP.