Blockchain systems based on the proof-of-work (PoW) consensus introduce entropy to the system in a natural way due to the randomness of mining. However, for non-PoW consensus (e.g., proof-of-stake and delegated proof-of-stake consensus) blockchain systems, a different approach to introducing entropy, such as the distributed random number generation (dRNG) algorithm, must be established. The dRNG algorithm is one of the key challenges in developing the consensus mechanism, as well as one of the relevant parameters for determining the merit of the consensus mechanism. In this paper, we first derive a publicly verifiable quantum random numbers generation protocol based on the certifiable randomness scheme from any untrusted quantum device, which offers features, such as fairness, no trusted third party, and publicly verifiable. Then, based on verifiable quantum random numbers, we propose a new consensus algorithm. The algorithm selects block proposer and block verification committees for each round using verifiable quantum random numbers, resulting in better randomness, fairness, and efficiency of the entire consensus process. In addition, the new consensus algorithm is not only resistant to adaptive adversary models as well as to collusion attacks, but also requires negligible computation for each user to avoid unnecessary consumption of power resources. Finally, we analyze the verifiable randomness, fairness, liveness, and communication complexity of the consensus algorithm.