In a recent work with Bolton Bailey (still not peer-reviewed) , we showed how a single quantum miner, with relatively little hashing power, can execute a 51% attack. The attack isn't relevant for the forthcoming years, requiring an extremely fast, noise-tolerant quantum computer.

The attack is surprisingly simple. The attacker creates a private fork, increasing the difficulty by a factor c. Due to the properties of Grover's algorithm, it is only \sqrt c harder for the quantum miner to mine at the new difficulty level, but these blocks count as $c$ times more for the PoW. Therefore, by mining even a single epoch for a large enough $c$, the quantum miner can generate more proof-of-work than the competing (classical) chain. The complexity of the attack is ~1/r^2 epochs, where r is the fraction of the block rewards that the quantum miner would have received if they mined honestly. This attack (or variants thereof) provides essentially the same benefits as classical 51% attacks, including double spending, and all the revenue from the block rewards.