Tightness of the entangled soundness bound and minimal quantum winning probability in 3-variable LCS games

Determine the minimal winning probability achievable by entangled quantum strategies in two-prover one-round linear constraint system games in which every equation involves exactly three variables (the class arising from the long-code test analyzed in this work), and ascertain whether the current soundness bound of 35/36 is tight; in particular, decide whether the minimal quantum winning probability coincides with the classical minimum of 5/6 or is strictly larger.

Background

The paper analyzes an entangled-prover variant of Håstad’s long-code test and derives an LCS game in which all equations have exactly three variables. For classical provers, any such system admits an assignment satisfying at least half the equations, yielding a tight classical lower bound of 5/6 for the game’s winning probability. In the quantum (entangled) setting, the authors obtain a soundness bound of 35/36.

The authors explicitly state uncertainty about the tightness of this entangled bound and pose the problem of determining the minimal winning probability for entangled strategies in this specific class of LCS games, together with whether it matches or exceeds the classical threshold. Resolving this would clarify the quantitative gap (if any) between classical and entangled strategies for these LCS games.