The statistics of a spectrum and its energy correlations define the presence or absence of chaotic behavior. This work sheds new light on spectral correlations and provides refined tools to diagnose quantum chaos quantitatively. Specifically, we study energy correlations between any two spectral levels; we provide the probability distribution of the kth energy neighbor and how this energy splitting contributes to the spectral form factor. We derive analytical results in the three Gaussian random matrix ensembles, which model quantum chaotic systems and for a Poissonian ensemble with independent energies, which models integrable systems, and illustrate how our results can be used to characterize a physical system—we choose the XXZ spin chain with disorder that can transition between the chaotic and integrable models. Thus, we can probe small deviations between the correlations shown by XXZ (in the chaotic and integrable phases) and the values expected from random matrix theory through Bohigas-Giannoni-Schmitt and Berry-Tabor conjectures. The most relevant deviations happen for the integrable phase, which shows considerably more correlations than those expected for Poisson, for which we give a qualitative explanation. Based on full-range spectral correlations, our refined probes of quantum chaos open a window to studying systems between the fully chaotic or fully integrable models, where there may be correlations between nearest neighbors but weaker correlations for further-away neighbors, or vice versa.