||In this paper, we investigate the efficiency in the art markets, using a generalized spectral test (GST) in a rolling window approach to detect departure from the martingale difference hypothesis (MDH) and trace the periods of market efficiency over time. Then we complement our results using the approximate entropy, the rescaled range analysis, and fractal dimension. We combine the three measures in an Efficiency Index for each market. Applying these methods, we find that the Modern Art, Paintings, Post-war, Prints, the USA market, and the global market in Euro show the largest values for the Approximate Entropy. Using the rescaled range estimates, we find that all markets are characterized by persistent behavior and, then using the Efficiency Index, our results indicate overwhelming evidence of market inefficiency in almost all sectors. Finally, we support our findings with some explanation of the reasons behind market inefficiency, related to asymmetrical information, influential galleries power, and differentiated pieces and talents in the art markets.