algorithm, but it is common to all Monte Carlo approaches to the calculation of quantum averages: the [*infamous*]{} sign-problem. The sign-problem is one of the major unsolved problems in the physics of quantum systems. Within the SSTP algorithm, it manifests itself both through the oscillating phase factors associated with the propagation on mean-energy surfaces and through the accumulation of fluctuating weights associated with the Monte Carlo sampling of the quantum transitions. In practice, upon analyzing the results obtained by means of this algorithm , we can conclude that the more quantum is the character of the bath the greater is the error in the calculation of the averages.

The mapping of the calculation of averages via the SSTP algorithm onto a stochastic process is reminiscent of the approach to open quantum system dynamics provided by the Stochastic Liouville Equation (SLE) [@sle1; @sle2; @sle3; @sle4]. However, in contrast to the SLE, the QCLE is a deterministic equation that explicitly takes into account all the DOF of the system without approximating the memory of the total hybrid quantum-classical system. The stochastic process only enters through the specific hybrid Molecular Dynamics/Monte Carlo implementation provided by the SSTP algorithm. Indeed, a recently proposed scheme 