PapersWe construct an explicit family of 3XOR instances which are hard for $O(\sqrt{\log n})$ levels of Sum-of-Squares. Our constructions are based on the LSV complexes, and rely on two of their crucial properties: cosystolic expansion and local nonpositive curvature (via Gromov’s filling inequality, which generalizes the isoperimetric inequality in $\mathbb{R}^n$)).
In contrast to many other constructions, our variables correspond to edges in the complex. Curiously, Alev, Jeronimo and Tulsiani showed that if variables correspond to vertices, then instances based on high-dimensional expanders are easy.
Using a different chain complex, Max Hopkins and Ting–Chun Lin were able to produce an instance which is hard for $\Omega(n)$ levels.
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