In the first part of talk, I will introduce a new quantum gravity observable, the de Sitter scattering amplitude defined in global de Sitter spacetime, which is non-perturbatively field basis independent and gauge invariant, a property that the conventional (A)dS/CFT observable fails to preserve. In de Sitter space, due to lack of time translation invariance, the scattering amplitude in general does not conserve energy. Remarkably, by imposing a notion of generalised energy conservation (GEC), we are able to uniquely pin down Dirac-Born-Infeld, Special Galileon and find evidence for new theories with higher mass spectrum. I also argue GEC serves as a stability condition for de Sitter vacua in quantum gravity.

In the second part of the talk, I will demonstrate that the non-energy conserving scattering amplitude precisely corresponds to the conformal anomaly of the boundary dual field theory. The GEC condition can therefore be translated into quantum conformal invariance of the boundary field theory. Consequently, by studying the boundary locality structure of the conformal anomaly, we find that holographic consistency lies in a strict measure-zero subset of GEC landscape.