  We study quantum field theory on a de Sitter spacetime dS$_{d+1}$ background.
Our main tool is the Hilbert space decomposition in irreducible unitary
representations of its isometry group $SO(d+1,1)$. As the first application of
the Hilbert space formalism, we recover the K\"allen-Lehmann spectral
decomposition of the scalar bulk two-point function. In the process, we exhibit
a relation between poles in the corresponding spectral densities and the
boundary CFT data. Moreover, we derive an inversion formula for the spectral
density through analytical continuation from the sphere and use it to find the
spectral decompisiton for a few examples. Next, we study the conformal partial
wave decomposition of the 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$_2$.
