We study Quantum Field Theory (QFT) on a background de Sitter spacetime dS_{d+1}. Our main tool is the Hilbert space decomposition in irreducible unitarity representations of its isometry group SO(d+1,1). Throughout this work, we focus on the late-time physics of dS_{d+1}, in particular on the boundary operators that appear in the late-time expansion of bulk local operators. As a first application of the Hilbert space formalism, we recover the Källen-Lehmann spectral decomposition of bulk two-point functions. In the process, we exhibit a relation between poles in the corresponding spectral densities and boundary CFT data. Next, we study the conformal partial wave decomposition of four-point functions of boundary operators. These correlation functions are very similar to the ones of standard conformal field theory, but have different positivity properties that follow from unitarity in de Sitter. We conclude by proposing a non-perturbative conformal bootstrap approach to the study of these late-time four-point functions, and we illustrate our proposal with a concrete example for QFT in dS_2.
