In the study of the capacity problem for multiple access channels (MACs), a lower bound on the error probability obtained by Han plays a crucial role in the converse parts of several kinds of channel coding theorems in the information-spectrum framework. Recently, Yagi and Oohama showed a tighter bound than the Han bound by means of Polyanskiy's converse. In this paper, we give a new bound which generalizes and strengthens the Yagi-Oohama bound, and demonstrate that the bound plays a fundamental role in deriving extensions of several known bounds. In particular, the Yagi-Oohama bound is generalized to two different directions; i.e, to general input distributions and to general encoders. In addition we extend these bounds to the quantum MACs and apply them to the converse problems for several information-spectrum settings.