On the Quantum Chromatic Gap II

Abstract

We confirm a conjecture of the first author [SODA'26] on a strong form of pseudo-telepathy for Khot's $d$-to-$1$ games: for every $d\geq 4$, every $ε>0$, and every $k\in\mathbb N$, there is a $d$-to-$1$ label-cover game of classical value less than $ε$ that admits a perfect quantum strategy using finite-dimensional shared entanglement with measurement contexts of radius at least $k$. As a consequence, there exist graphs whose quantum chromatic number $χ_q$ is $3$ and whose classical chromatic number $χ$ is arbitrarily large. This is the first improvement on the quantum chromatic gap since the work of Brassard, Cleve, and Tapp [Phys. Rev. Lett.'99], which produced graphs for which $χ$ grows exponentially in $χ_q$. To prove the conjecture, we proceed as follows. We start with a ``stretched'' version of the Mermin--Peres magic square, obtained by spacing out its equations and linking them with additional long equation wires. This operation increases the radius of the initial measurement contexts of a perfect quantum strategy for the game, while preserving classical unsatisfiability. We then apply parallel repetition to make the classical value small, and we turn the system into a pseudo-telepathic multi-layered Kneser hypergraph via a PCP construction of Dinur, Guruswami, Khot, and Regev [STOC'03]. Finally, we feed the resulting hypergraph into the reduction of Fei, Minzer, and Wang from hypergraph colouring to $4$-to-$1$ games, we establish quantum completeness of this reduction, and we show that it preserves context radius.