Equality is Far Weaker than Constant-Cost Communication

Abstract: We exhibit an $n$-bit communication problem with a constant-cost randomized protocol but which requires $n^{\Omega(1)}$ deterministic (or even non-deterministic) queries to an Equality oracle. Therefore, even constant-cost randomized protocols cannot be efficiently "derandomized" using Equality oracles. This improves on several recent results and answers a question from the survey of Hatami and Hatami (SIGACT News 2024). It also gives a significantly simpler and quantitatively superior proof of the main result of Fang, G\"o\"os, Harms, and Hatami ( STOC 2025), that constant-cost communication does not reduce to the $k$-Hamming Distance hierarchy.