In CTTSH the force for the propagation of the nuclear trajectories is the same as in TSH, whereas CTTSH includes an additional term in the propagation of electronic coefficients that depends on quantum momentum, as shown in Eq. (17). TSH is known to suffer from a systematic issue termed "overcoherence", arising from the disconnect between the evolution of electrons and nuclei. Throughout time, the nuclei evolve on a single adiabatic potential energy surface and are allowed to hop to another surface at any time based on a stochastic algorithm. Therefore, on the one hand, the nuclear dynamics is governed by a single electronic states, while, on the other hand, the electronic evolution remains always coherent along a trajectory and yields a superposition of adiabatic states along that trajectory. CTTSH, thanks to the coupled-trajectory term of Eq. ( ), improves the description of quantum decoherence compared to TSH as it accounts in the electronic evolution equation for information about the spatial delocalization of the trajectories. This information is encoded in quantum momentum (14), which is related to the spatial variation of the nuclear density. Then, the nuclear density needs to be constructed from the distribution of trajectories, and this is done in CTTSH, as in CTMQC, with a sum of normalized Gaussians centered at the positions of trajectories. Therefore, at a given time t, the positions of all the trajectories need to be "shared" among the trajectories to construct the nuclear density and, thus, the trajectories cannot be evolved independently. From a practical standpoint, this is the main drawback of CTTSH when compared to standard TSH. For an efficient performance of CTTSH, a parallelized implementation using the Message Passing Interface (MPI) was employed in Ref. [199]. Despite the additional computation required for the quantum momentum, the runtime of our parallelized implementation is similar to that of a standard TSH approach, as all other calculations are performed in parallel. Furthermore, a local diabatization algorithm was utilized within the framework of the Runge-Kutta integrator to bypass the computation of the expensive and sometimes unavailable nonadiabatic coupling vectors.