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Learning about physical systems from quantumenhanced experiments, relying on a quantum memory and quantum processing, can outperform learning from experiments in which only classical memory and processing are available. Whereas quantum advantages have been established for a variety of state learning tasks, quantum process learning allows for comparable advantages only with a careful problem formulation and is less understood. We establish an exponential quantum advantage for (a) learning the Pauli transfer matrix of an arbitrary unknown nqubit quantum channel, (b) predicting expectation values of bounded Paulisparse observables measured on the output of an arbitrary channel upon input of a Paulisparse state, and (c) predicting expectation values of arbitrary bounded observables measured on the output of an unknown channel with sparse Pauli transfer matrix upon input of an arbitrary state. With quantum memory, these tasks can be solved using linearlyinn many copies of the Choi state of the channel, and even timeefficiently in the case of (b). In contrast, any learner without quantum memory requires exponentiallyinn many queries, even when querying the channel on subsystems of adaptively chosen states and performing adaptively chosen measurements. In proving this separation, we extend existing shadow tomography upper and lower bounds from states to channels via the ChoiJamiolkowski isomorphism. Moreover, we combine Pauli transfer matrix learning with polynomial interpolation techniques to develop a procedure for learning arbitrary Hamiltonians, which may have nonlocal alltoall interactions, from shorttime dynamics. Our results highlight the power of quantumenhanced experiments for learning highly complex quantum dynamics. Host: Frederic Sauvage (T4) 