As a distributed network, the operational efficacy of a blockchain system relies heavily on the consensus mechanism it adopts. Among the mainstream consensus mechanisms in blockchain, proof-of-authority (PoA) is appealing for its post-quantum security and block generation efficiency and, therefore, has gained academic attention. Nevertheless, the traditional PoA faces three major problems: (1) Low decentralization. The allocation of bookkeeping rights has a low degree of decentralization in the blockchain network. (2) Low availability. The total exposure of the leader node makes it susceptible to centralized attacks, leading to a single point of failure (SPOF) dilemma and reducing the availability of the entire blockchain system. (3) Non-robustness (low fault tolerance). Generating a new block must be done jointly by all validator nodes, with this voting process potentially delayed due to invalid voter participation. To address these issues, this paper improves PoA and proposes a new consensus mechanism scheme, the random quantum proof of authority (RQPoA). First, RQPoA develops the leader node election algorithm with a verifiable delay function (VDF) to realize fair and impartial leader node selection, enforcing the blockchain’s decentralization level. Second, RQPoA adopts the multi-party quantum secret sharing protocol to share the leader node’s identity among validator nodes confidentially. This strategy eliminates SPOF caused by the leader node, increasing the blockchain system availability. Third, RQPoA incorporates a candidate block voting protocol based on a quantum threshold signature to complete the block proposal, which is fault-tolerant and thus enhances the blockchain system's robustness. A security analysis of RQPoA demonstrates its security, efficiency, and better fault tolerance than related quantum consensus mechanisms. In conclusion, the RQPoA makes a useful exploration for researching secure consensus mechanisms in the post-quantum era and enriches the related research.