The discovery and the ensuing burst of applications of layered nanomaterials, including single-layer and multilayer graphene, hexagonal boron nitride, and transition metal dichalcogenides have undoubtedly revolutionized materials science, revealing bright prospects in nanotechnology and other related fields. Low-dimensional nanostructures have been demonstrated to possess previously unexpected electronic, optical, cohesive, and thermal properties and are widely used as an important part of functional materials and surfaces. In particular, atomically-thin two-dimensional materials such as graphene and hexagonal boron nitride have recently been found to exhibit appreciable permeability to thermal protons, making them emerging candidates for separation technologies [S. Hu et al., Nature 516, 227 (2014); M. Lozada-Hidalgo et al., Science 351, 68 (2016)]. These remarkable experimental observations extend the appeal of two-dimensional materials for a wide range of applications ranging from water desalination to exchange membranes in fuel cells or sensors. However, the high experimentally observed permeability of pristine graphene to hydrogen ions remains puzzling for first-principles quantum-mechanical calculation, because it cannot be explained within electronic-structure calculations. An observed surface areal conductivity difference for protons and deuterons also demonstrates the crucial importance of nuclear quantum effects (NQE) in this process.The QUANTION project will convincingly elucidate the role of NQE for proton tunneling and scattering in layered nanomaterials, such as graphene, hBN, MoS2, phosphorene, MoSe2, WS2, and SnS2. Due to strong interatomic interactions within nanomaterials, NQE may substantially modify the stability and thermodynamic properties of nanomaterials and qualitatively change the mechanisms of processes involving interactions with atomic and molecular species. The quantum delocalization of nuclei can considerably affect their mobility, play an important role in adsorption processes, and is essential for the description of many dynamical phenomena. However, our knowledge about of the interplay of quantum-mechanical electronic and nuclear effects for such complex systems is in its infancy, thus advancing the understanding of such phenomena is one of the main goals of this proposal. Our research will not only shed light on the underlying quantum mechanisms, but also will provide new insights for rational design of highly efficient functional materials with controllable properties based on quantum-mechanical principles.