The role of the carbon market in relation to the cryptocurrency market is still unclear. Given the carbon-intensive nature of the cryptocurrency industry, whether the carbon market is able to capture the carbon footprint of the cryptocurrency market (i.e., diversification) or act as a safe haven or a hedge against it remain unexplored issues. To address this issue, this paper employs the generalized autoregressive score-dynamic conditional score-Copula (GAS–DCS–Copula) model, incorporating the asymmetric tail distribution. We identify the asymmetric tail properties of both the carbon and cryptocurrency markets with significant otherness. Further, to account the importance of China in mining the cryptocurrencies, we incorporate Chinese carbon market in our analysis to investigate the difference with the European carbon market. Finally, we provide evidence that the European carbon market provides a safe haven and a hedge against the cryptocurrency market while Chinese carbon market is not. Our findings have implications for both investors and policymakers.