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The study of non-locality is fundamental to the understanding of quantummechanics. The past 50 years have seen a number of non-locality proofs, but itsfundamental building blocks, and the exact role it plays in quantum protocols,has remained elusive. In this paper, we focus on a particular flavour ofnon-locality, generalising Mermin's argument on the GHZ state. Using stronglycomplementary observables, we provide necessary and sufficient conditions forMermin non-locality in abstract process theories. We show that the existence ofmore phases than classical points (aka eigenstates) is not sufficient, and thatthe key to Mermin non-locality lies in the presence of certain algebraicallynon-trivial phases. This allows us to show that fRel, a favourite toy model forcategorical quantum mechanics, is Mermin local. We show Mermin non-locality tobe the key resource ensuring the device-independent security of the HBB CQ(N,N) family of Quantum Secret Sharing protocols. Finally, we challenge theunspoken assumption that the measurements involved in Mermin-type scenariosshould be complementary (like the pair X,Y), opening the doors to a much widerclass of potential experimental setups than currently employed. In short, wegive conditions for Mermin non-locality tests on any number of systems, whereeach party has an arbitrary number of measurement choices, where eachmeasurement has an arbitrary number of outcomes and further, that works in anyabstract process theory.

Mermin Non-Locality in Abstract Process Theories