What Can Be Observed Locally?

We consider the question of locality in distributed computing in the context of quantum information. Specifically, we focus on the round complexity of quantum distributed algorithms, with no bounds imposed on local computational power or on the bit size of messages. Linial's LOCAL model of a distributed system is augmented through two types of quantum extensions: (1) initialization of the system in a quantum entangled state, and/or (2) application of quantum communication channels. For both types of extensions, we discuss proof-of-concept examples of distributed problems whose round complexity is in fact reduced through genuinely quantum effects. Nevertheless, we show that even such quantum variants of the LOCAL model have non-trivial limitations, captured by a very simple (purely probabilistic) notion which we call "physical locality" (φ-LOCAL). While this is strictly weaker than the "computational locality" of the classical LOCAL model, it nevertheless leads to a generic view-based analysis technique for constructing lower bounds on round complexity. It turns out that the best currently known lower time bounds for many distributed combinatorial optimization problems, such as Maximal Independent Set, bounds cannot be broken by applying quantum processing, in any conceivable way.