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We exhibit a total Boolean function on n input bits whose bounded-error quantum simultaneous-message (and hence one-way) communication complexity is in O(√ n), whereas its randomised one-way (and hence simultaneous-message) communication complexity is in Ω(n). The same function admits a classical simultaneous-message protocol of cost O(√ n) when the players share O(√ n) EPR pairs.
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Dmytro Gavinsky (2026) studied this question.
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