This study analyzes how the 2008 and 2010 financial crises, which began in the US and Greece respectively, affected the Hurst exponents of index returns of the stock markets of Belgium, France, Greece, Japan, the Netherlands, Portugal, the UK and US. We perform two innovative statistical tests for this purpose. The first assesses whether the returns exhibit a long memory in the pre-crisis and crisis periods and determines the extent to which the Hurst exponents, calculated with the multifractal detrended moving average technique (MFDMA), differ from the tranquil to the crisis periods. The second test uses copula models to assess whether the correlation between the local Hurst exponents of the markets where the crises originated and those of the other markets increased due to the crises. The results of the first test suggest that although most of the returns exhibit a long memory in the 2008 crisis period, this is not the case in either the pre-crisis or the 2010 crisis periods. These findings shed light on the dynamics of market efficiency. The results of the second test show a significant increase in correlation between the local Hurst exponents of several markets, suggesting the existence of financial contagion. We observed that the 2008 crisis had a greater impact on the memory properties of stock returns than the 2010 financial crisis.