It is becoming a common wisdom that a full-fledge formulation of quantum gravity needs to prescind from at least one of the two pillars on which our current understanding of nature is based, i.e. locality and unitarity. In order to get a clue of how this particular regime is reached, it is appropriate to reformulate the physics we know, which indeed is both local and unitary, in such a way that neither locality nor unitarity are fundamental assumptions but rather they are emergent. Small steps in this direction have been done in the area of scattering amplitudes for $N=4$,  where locality and unitarity are tied to the positivity of the geometry of some polytope through which the theory can be formulated.

In this talk, I will report on some very recent progress towards a similar formulation for correlators, which holds not only in flat space but also in a cosmological set-up (dS) or in anti-de-Sitter space. Interestingly, new mathematical structures arise. I will focus on new perturbative representations for such correlators and their interpretation in terms of new polytopes.