The electricity transactions of microgrids face several problems: the high platform management cost, the low security, and the untimely consumption of scattered electricity. To solve these problems, this paper presents a multi-microgrid thermal game model based on quantum blockchain. Specifically, a dynamic model was established for the noncooperative game between aggregators, microgrids, and large users to maximize the benefit of each party, and to realize the timely consumption of scattered electricity. Next, a transaction platform was constructed based on the two-round password based authenticated key exchange (PAKE) protocol, which eliminates non-interactive zero-knowledge (NIZK), aiming to substantially enhance the post-quantum security of transactions. Then, the quantum signature using two-particle entangled Bell states was adopted to safeguard the quantum communication of electricity transactions, and authenticate the nodes. Example analysis shows that our model can realize the timely consumption of scattered electricity and thermal energy, improve the security of transaction data and users, and achieve Pareto optimality. The research provides theoretical support and decision-making basis for electricity transactions in the post-quantum age.