# The SDP value for random two-eigenvalue CSPs

## TL;DR

This paper precisely determines the SDP (quantum) value of large random two-eigenvalue CSPs, extending previous results and introducing new techniques that may indicate a computational threshold.

## Contribution

It extends the analysis of SDP values to new classes of random CSPs using advanced spectral and nonbacktracking operator techniques.

## Key findings

- SDP value matches spectral relaxation in large random instances
- Includes new cases like random Sort4 and Forrelation CSPs
- Techniques generalize nonbacktracking operators and Ihara--Bass Formula

## Abstract

We precisely determine the SDP value (equivalently, quantum value) of large random instances of certain kinds of constraint satisfaction problems, ``two-eigenvalue 2CSPs''. We show this SDP value coincides with the spectral relaxation value, possibly indicating a computational threshold. Our analysis extends the previously resolved cases of random regular $\mathsf{2XOR}$ and $\textsf{NAE-3SAT}$, and includes new cases such as random $\mathsf{Sort}_4$ (equivalently, $\mathsf{CHSH}$) and $\mathsf{Forrelation}$ CSPs. Our techniques include new generalizations of the nonbacktracking operator, the Ihara--Bass Formula, and the Friedman/Bordenave proof of Alon's Conjecture.