By mapping out potential energy surfaces from density-functional theory (DFT) and solving a protonic Schrodinger equation, we find that the H atom in a unit cell of the Li2NH crystal shows remarkably strong quantum behavior, leading to the delocalization of H over six octahedral sites around each N. This can be rationalized in terms of rapid coherent tunneling among these equivalent octahedral sites. Structural and dynamical consequences of our finding are discussed. Since the Li-N-H compounds are considered promising candidates for H-storage, understanding of these fundamental properties will be useful toward improving the performance of the material.