We propose a new strategy to constraint (and perhaps determine) the mass spectrum and the scattering amplitudes of a Quantum Field Theory (QFT). We consider QFT in hyperbolic space and study correlation functions of operators inserted at the conformal boundary. By construction, these observables transform like correlation functions of a lower dimensional Conformal Field Theory. We then apply conformal bootstrap techniques to find universal bounds on the mass spectrum and scattering amplitudes of the QFT. For 1+1 dimensional QFT, we recover the same bounds from a direct S-matrix bootstrap approach.