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DOI: 10.4230/LIPIcs.ITCS.2026.113

Multi-Quadratic Sum-Of-Squares Lower Bounds Imply VNC ¹ ≠ VNP

Authors: Benjamin Rossman and Davidson Zhu

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract

The 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 = ∑_{i = 1}^s g_i² with each g_i d-multilinear. In the case d = 2, Hrubeš, Wigderson and Yehudayoff [Hrubeš et al., 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 multiquadratic sum-of-squares and 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) ≤ d(n) ≤ O((log n)/(log log n)), would separate the algebraic complexity classes VNC¹ and VNP.

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Benjamin Rossman and Davidson Zhu. Multi-Quadratic Sum-Of-Squares Lower Bounds Imply VNC ¹ ≠ VNP. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 113:1-113:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{rossman_et_al:LIPIcs.ITCS.2026.113,
  author =	{Rossman, Benjamin and Zhu, Davidson},
  title =	{{Multi-Quadratic Sum-Of-Squares Lower Bounds Imply VNC ¹ ≠ VNP}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{113:1--113:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.113},
  URN =		{urn:nbn:de:0030-drops-254006},
  doi =		{10.4230/LIPIcs.ITCS.2026.113},
  annote =	{Keywords: sum-of-squares, arithmetic formulas}
}

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Multi-Quadratic Sum-Of-Squares Lower Bounds Imply VNC ¹ ≠ VNP

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