They propose a scenario where a quantum miner, even with significantly less hashing power compared to traditional miners, could feasibly execute a 51% attack on a blockchain network. This form of attack, however, is not an immediate concern as it necessitates the availability of an advanced, noise-tolerant quantum computer capable of operating at speeds far beyond current technological capabilities.

The mechanism of the attack is described as straightforward yet effective, hinging on the exploitation of Grover's algorithm—a quantum algorithm that provides quadratic speedup for search problems. By initiating a private fork and artificially inflating the mining difficulty, a quantum miner can utilize Grover's algorithm to mine blocks at this elevated difficulty with relatively lesser effort (only (\sqrt{c}) times harder) than would be required by classical computational means. Despite the increase in difficulty, each block mined in this manner disproportionately contributes to the proof-of-work (PoW) consensus mechanism, allowing the attacker to potentially dominate the network's PoW with fewer blocks mined.

This approach directly correlates the success rate of the attack with the choice of the difficulty increase factor (c) and inversely with the square of the epochs ((1/r^2)), where (r) represents the proportion of block rewards the quantum miner would have received under honest mining practices. The implications of such an attack are profound, echoing the ramifications of traditional 51% attacks, such as the ability to double-spend and claim all block rewards, thus undermining the integrity and security of the blockchain.

Given the potential future relevance of this attack vector, especially in light of ongoing advancements in quantum computing and protocol evolution within blockchain technologies, the research underscores the need for preemptive consideration and possibly adaptation of blockchain protocols to mitigate against such quantum-enabled threats.