From correlators for all Lambdas to EAdS, and back

I will review some recent and on-going efforts to study holographic correlation functions in de Sitter (dS) and Minkowski space. After a brief review of AdS/CFT I will start reviewing the dS story, where in the Bunch-Davies vacuum one can perturbatively rewrite correlators on the future boundary in terms of Witten diagrams in Euclidean AdS upon considering appropriate Wick rotations. I will then explain how an analogous relation holds between celestial correlation functions and Witten diagrams. Contact diagram processes are proportional to contact Witten diagrams and particle exchanges can be recast as sums, discrete or continous, of particle exchanges in EAdS. Using such relations one can then import familiar EAdS techniques to study the properties of dS and celestial correlators. In particular, we infer the analytic structure of the spectral density in the conformal partial wave expansion of correlators finding that, at least at any order in perturbation theory, it is a meromorphic function of the energy. We conclude discussing non perturbative constraints from unitarity in Euclidean Conformal Field Theory.