In this paper, we describe a new method for simulating quantum processes using classical-like molecular dynamics. The approach is based on solving the quantum Liouville equation in the Wigner representation using ensembles of classical trajectories in phase space. The nonlocality of quantum mechanics is incorporated in the trajectory representation as nonclassical interactions between the members of the ensemble, leading to an entanglement of their evolution. The statistical independence of the individual trajectories making up an ensemble in the classical limit is lost when quantum effects are included, and the entire state of the system must be propagated as a unified whole. We develop the formalism and its numerical implementation, and illustrate its application on two model problems of quantum mechanical tunneling: escape from a metastable well and wave packet penetration of the Eckhart barrier. (C) 2003 American Institute of Physics.