Multiquadratic Sum-of-Squares Lower Bounds Imply VNC$^1$ $\neq$ VNP

TL;DR

This paper links lower bounds on the sum-of-squares complexity of multiquadratic polynomials to major complexity class separations, suggesting that proving such bounds could resolve fundamental questions in algebraic complexity theory.

Contribution

It establishes a novel connection between sum-of-squares lower bounds for multiquadratic polynomials and the separation of VNC$^1$ and VNP complexity classes.

Findings

Lower bounds on SoS complexity imply class separations.

Extends known biquadratic results to multiquadratic case.

Provides a new approach to algebraic complexity lower bounds.

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 $d=2$, Hrube\v{s}, 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.