  Understanding how the interplay between higher-order and multilayer
structures of interconnections influences the synchronization behaviors of
dynamical systems is a feasible problem of interest, with possible application
in essential topics such as neuronal dynamics. Here, we provide a comprehensive
approach for analyzing the stability of the complete synchronization state in
simplicial complexes with numerous interaction layers. We show that the
synchronization state exists as an invariant solution and derive the necessary
condition for a stable synchronization state in presence of general coupling
functions. It generalizes the well-known master stability function scheme to
the higher-order structures with multiple interaction layers. We verify our
theoretical results by employing them on networks of paradigmatic R\"{o}ssler
oscillators and Sherman neuronal models, and demonstrate that the presence of
group interactions considerably improves the synchronization phenomenon in the
multilayer framework.
