Multiquadratic Sum-of-Squares Lower Bounds Imply VNC$^1$ $\neq$ VNP
Benjamin Rossman, Davidson Zhu
TL;DR
This work links strong lower bounds on the SoS complexity of d-multiquadratic polynomials to separations in algebraic complexity, notably suggesting that n^{d−o(log d)} SoS lower bounds would separate VNC^1 from VNP for suitably growing d. Central to the approach is the PT-rank, a generalized partial transpose-based measure that tightly connects SoS and multiquadratic forms to set-multilinear formulas and to both non-commutative and commutative formula sizes. The authors establish a main lower-bound theorem: L_sm(ShiftedTensor(M)) ≥ PT-rank(M)/n^{d−log d+1}, leading to VNC^1 ≠ VNP under explicit PT-rank assumptions for M with d in the stated range. They also provide upper bounds and explicit candidate matrices, clarifying the landscape of potential hard instances and signaling both the promise and the limitations of PT-rank as a route to strong algebraic 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 $d=2$, 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.