We exhibit an n-bit communication problem with a constant-cost randomized protocol but which requires nΩ(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.
Cite
@article{arxiv.2507.11162,
title = {Equality is Far Weaker than Constant-Cost Communication},
author = {Mika Göös and Nathaniel Harms and Artur Riazanov},
journal= {arXiv preprint arXiv:2507.11162},
year = {2025}
}