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We explore a connection between monogamy of non-locality and a weakmacroscopic locality condition: the locality of the average behaviour. Theseare revealed by our analysis as being two sides of the same coin.  Moreover, we exhibit a structural reason for both in the case of Bell-typemultipartite scenarios, shedding light on but also generalising the results inthe literature [Ramanathan et al., Phys. Rev. Lett. 107, 060405 (2001);Pawlowski & Brukner, Phys. Rev. Lett. 102, 030403 (2009)]. More specifically,we show that, provided the number of particles in each site is large enoughcompared to the number of allowed measurement settings, and whatever themicroscopic state of the system, the macroscopic average behaviour is localrealistic, or equivalently, general multipartite monogamy relations hold.  This result relies on a classical mathematical theorem by Vorob'ev [TheoryProbab. Appl. 7(2), 147-163 (1962)] about extending compatible families ofprobability distributions defined on the faces of a simplicial complex -- inthe language of the sheaf-theoretic framework of Abramsky & Brandenburger [NewJ. Phys. 13, 113036 (2011)], such families correspond to no-signallingempirical models, and the existence of an extension corresponds to locality ornon-contextuality. Since Vorob'ev's theorem depends solely on the structure ofthe simplicial complex, which encodes the compatibility of the measurements,and not on the specific probability distributions (i.e. the empirical models),our result about monogamy relations and locality of macroscopic averages holdsnot just for quantum theory, but for any empirical model satisfying theno-signalling condition.  In this extended abstract, we illustrate our approach by working out a coupleof examples, which convey the intuition behind our analysis while keeping thediscussion at an elementary level.

On monogamy of non-locality and macroscopic averages: examples and preliminary results