The purpose of paper is to assess the long-term memory of stock index returns in the pan-European platform Euronext (CAC-40, AEX, BEL-20 and PSI-20). We find evidence of time dependency in much of the data, suggesting that the series may best be described as fractional Brownian motion. Modified Rescaled-Range Analysis and Detrended Fluctuation Analysis were used to measure the degree of long memory. The global Hurst exponents evidence persistent long memory in the Dutch, Belgian and Portuguese markets. In the French market, evidence of long memory is inconsistent and weak. Fractal structure suggests non-conformity with the Efficient Market Hypothesis, and may compromise the reliability of asset pricing models. Furthermore, time-dependent Hurst exponents show evidence of weakening persistence in these markets, particularly after the international crises of 2000, 2002 and 2010. A possible explanation for those changes is that the markets may have matured over time, becoming more efficient after these severe events.