This study has investigated the changing efficiency for the phase III EU ETS CO 2 market using the daily historical data of allowance future prices and coverage from August 2015 to December 2020. We have applied two alternative tests for checking dependency by linear and nonlinear methods to achieve this goal, including generalized spectral (GS) and automatic portmanteau (AQ). Also, we had a comprehensive look at the carbon market evolution and the EU ETS scheme development over time. The analysis of observed results validates the adaptive market hypothesis (AMH) in the market, which corresponds with the oscillatory behavior of the applied test statistics' p-values. The other aspect of the study was to analyze the existence of evolutionary behavior on the market. To reach this purpose, we checked the results by applying a rolling window technique with four different time windows (50, 100, 150, and 250 days) on the test statistics in harmony with the adaptive market hypothesis. The obtained results show that overall, market efficiency has been improved by implementing higher window lengths.