On account of the recent advances in quantum computers, it becomes pressing to consider quantum-safe authentication schemes for two-way communications in neighborhood area networks of smart grid, i.e., preventing possible attacks that the quantum computers could bring to the grid. In this paper, we take a further step toward this goal by integrating the lattice-based public-key encryption and key exchange techniques to provide mutual authentication between smart meters and the neighborhood gateway. Security analysis shows that our proposed scheme is secure under attacks from both quantum and conventional computers, including the recently introduced key reuse attacks. In addition, the proposed schemes can also achieve forward security, which guarantees that the secrets leaked in the current session will not result in the leakage of secrets in former sessions. Furthermore, the proposed scheme also gives prior consideration to the smart meters with limited computation capacity and puts as little burden on them as possible. Extensive performance analysis is also conducted, and the results demonstrate the efficiency of our proposed scheme, as well as the relatively low communication and storage costs.