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Sign-rank of the Hypercube matrix grows unboundedly

Prove that the sign-rank of the Hypercube sign matrix Q_n, defined by Q_n(x,y) = −1 iff x and y differ in exactly one coordinate, satisfies lim_{n→∞} signrank(Q_n) = ∞.

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Background

The Hypercube problem Q_n has constant-cost randomized protocols (Q_n ∈ BPP_0), yet its sign-rank is unknown. An unbounded sign-rank would imply Q_n ∉ UPP_0, separating BPP_0 and UPP_0 and yielding new insights into the power of randomness vs. unbounded-error models.

Because wrect(Q_n){-1} = O(1), proving this conjecture likely requires new techniques beyond rectangle and margin/VC methods.

References

We conjecture that the sign-rank of $Q_n$ tends to infinity as $n$ grows, which, if true, would imply $BPP_0 \not\subseteq UPP_0$. \begin{conjecture}[Sign-rank of Hypercube] We have \lim_{n \to \infty} _\pm(Q_n)=\infty. \end{conjecture}

— Structure in Communication Complexity and Constant-Cost Complexity Classes (2401.14623 - Hatami et al., 26 Jan 2024) in Conjecture [Sign-rank of Hypercube], Section 5.3 (Sign-rank and UPP)