A unique feature of ultracold atoms is the separation of length scales, r_{0}≪k_{F}^{-1}, where k_{F} and r_{0} are the Fermi momentum characterizing the average particle distance and the range of interaction between atoms, respectively. For s-wave scattering, Shina Tan discovered that such diluteness leads to universal thermodynamic relations governed by contact. Here, we show that the concept of contact can be generalized to an arbitrary partial-wave scattering. Contact of all partial-wave scatterings forms a contact spectrum, which establishes universal thermodynamic relations with notable differences from those in the presence of s-wave scattering alone. Such a contact spectrum is particularly useful for characterizing many-body correlations in atomic quantum Hall states (QHSs). It has an interesting connection with a special bipartite entanglement spectrum of QHSs and enables an intrinsic probe of atomic QHSs using short-range two-body correlations.