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Separation of bounded-error and unbounded-error randomized communication

Construct a two-party communication problem M whose public-coin bounded-error randomized communication complexity is constant (Rand(M)=O(1)) while its private-coin unbounded-error communication complexity grows super-constantly (U(M)=ω(1)).

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Background

The authors discuss a long-standing conjecture proposing a qualitative separation between bounded-error and unbounded-error randomized communication. Their main results rule out prominent candidate families (notably k-Hamming Distance and various compositions/reductions thereof), but the general separation conjecture is not resolved.

Establishing such a separation would settle whether constant-cost bounded-error randomized protocols necessarily imply constant-cost unbounded-error protocols, a question tightly linked to the margin versus sign-rank problem.

References

There exists a communication problem M with \Rand(M) = O(1) but \U(M) = \omega(1).

— Sign-Rank of $k$-Hamming Distance is Constant (2506.12022 - Göös et al., 1 May 2025) in Section 1.2 (Sign-rank in communication), Boxconjecture (conj:intro-main)