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The complexity class ZPP NP[1] (corresponding to zero-error randomized algorithms with access to one NP oracle query) is known to have a number of curious properties. We further explore this class in the settings of time complexity, query complexity, and communication complexity. • For starters, we provide a new characterization: ZPP NP[1] equals the restriction of BPP NP[1] where the algorithm is only allowed to err when it forgoes the opportunity to make an NP oracle query. • Using the above characterization, we prove a query-to-communication lifting theorem , which translates any ZPP NP[1] decision tree lower bound for a function f into a ZPP NP[1] communication lower bound for a two-party version of f . • As an application, we use the above lifting theorem to prove that the ZPP NP[1] communication lower bound technique introduced by Göös, Pitassi, and Watson (ICALP 2016) is not tight. We also provide a “primal” characterization of this lower bound technique as a complexity class. 

Suppose Alice and Bob each start with private randomness and no other input, and they wish to engage in a protocol in which Alice ends up with a set x ⊆ [ n ] and Bob ends up with a set y ⊆ [ n ], such that ( x , y ) is uniformly distributed over all pairs of disjoint sets. We prove that for some constant β < 1, this requires Ω ( n ) communication even to get within statistical distance 1− β n of the target distribution. Previously, Ambainis, Schulman, Ta-Shma, Vazirani, and Wigderson (FOCS 1998) proved that Ω (√ n ) communication is required to get within some constant statistical distance ɛ > 0 of the uniform distribution over all pairs of disjoint sets of size √ n .