Title: Maximal Correlation-Based Post-Nonlinear Learning for Bivariate Causal Discovery. Abstract: Bivariate causal discovery aims to determine the causal relationship between two random variables from passive observational data (as intervention is not affordable in many scientific fields), which is considered fundamental and challenging. Designing algorithms based on the post-nonlinear (PNL) model has aroused much attention for its generality. However, the state-of-the-art (SOTA) PNL-based algorithms involve highly non-convex objectives for neural network training, which are time-consuming and unable to produce meaningful solutions with finite samples. In this paper, we propose a novel method that incorporates maximal correlation into the PNL model learning (short as MC-PNL) such that the underlying nonlinearities can be accurately recovered. Owing to the benign structure of our objective function when modeling the nonlinearities with linear combinations of random Fourier features, the target optimization problem can be solved rather efficiently and rapidly via the block coordinate descent. We also compare the MC-PNL with SOTA methods on the downstream synthetic and real causal discovery tasks to show its superiority in time and accuracy. Our code is available at https://anonymous.4open.science/r/MC-PNL-E446/.