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We study three prominent diagnostics of chaos and scrambling in the contextof two-dimensional conformal field theory: the spectral form factor,out-of-time-ordered correlators, and unitary operator entanglement. With theobservation that all three quantities may be obtained by different analyticcontinuations of the torus partition function, we address the connections anddistinctions between the information that each quantity provides us. In thisprocess, we study the emergence of irrationality from "large-N" limits ofrational conformal field theories (RCFTs) as well as the explicit breakdown ofrationality for theories with central charges greater than the number of theirconserved currents. Our analysis begins to elucidate the intermediate dynamicalbehavior of theories that bridge the gap between integrable RCFTs and maximallychaotic holographic CFTs.

Conformal field theory and the web of quantum chaos diagnostics