Abstract: Causal inference using observational data is challenging, especially in the bivariate case. Through the minimum description length principle, we link the postulate of independence between the generating mechanisms of the cause and of the effect given the cause to quantile regression. Based on this theory, we develop Quantile Causal Discovery (QCD), a new method to uncover causal relationships. Because it uses multiple quantile levels instead of the conditional mean only, QCD is adaptive not only to additive, but also to multiplicative or even location-scale generating mechanisms. To illustrate the empirical effectiveness of our approach, we perform an extensive empirical comparison on both synthetic and real datasets. This study shows that QCD is robust across different implementations of the method (i.e., the quantile regression algorithm), computationally efficient, and compares favorably to state-of-the-art methods.