As remarked by Max Jammer, there was rarely in the history of physics a comprehensive theory which owed so much to one principle as quantum mechanics owed to Bohr's correspondence principle. In this talk we comment on this principle in connection to two important quantities: entanglement and the out-of-time-ordered correlator (OTOC), a diagnostic tool which reduces to the Lyapunov exponent in the classical limit but is well defined for general quantum systems. The correspondence principle requires transition to classical mechanics for all quantities including dynamical chaos. Indeed, one of the main goals in the field of quantum chaos is to establish a correspondence between the dynamics of classical chaotic systems and their quantum counterparts. On the other hand the notion of entanglement is meaningful only in quantum mechanics and its relation to classical chaos is not obvious. In addition, the validity of the correspondence principle has been recently challenged by the observation of an early-time exponential increase of OTOC in classically nonchaotic systems. The role of OTOC as a diagnostic of chaos has also been put in question. Here, we show that the correspondence principle maintains its general validity while OTOC also maintains its role as a diagnostic of chaotic dynamics.