As a result of the gauge/gravity duality, the spectra of thermal correlation functions in certain quantum field theories equal the quasinormal mode (QNM) spectra of black holes. In my talk, I will discuss three facts about such large-N quantum field theories that can help us understand generic analytic properties of thermal correlators and answer the question of what is the minimal amount of information needed to uniquely determine correlation function. In the first part, I will discuss how and when the entire spectrum can be determined from the dispersion relation of a single QNM. In the second part, I will show how the phenomenon of pole skipping and an expansion around an infinite number of dimensions can be used to constrain and fix such spectra. Finally, in the third part, I will show a new product formula that follows from the S-duality and constrains the spectra of 3d CFTs with 4d bulk duals.