With the help of QM/EFP-MD with modern correlated quantum theories, distinctly different proton transport dynamics for hydronium and hydroxide ions was revealed. The efficiency of proton transfer for hydronium was found to be significantly higher than that for hydroxide, and the difference in efficiency increased as the temperature was lowered. This difference in dynamics suggests that molecular Brownian diffusion may play an important role in hydroxide transport. Our theoretical findings are consistent with recent experimental observations of proton transfer in amorphous solid water.