We study the impact that two miners equipped with quantum computers purpose-built for quantum Bitcoin mining will have on the 51% attack threshold of the Bitcoin network, given that the miners are playing a competitive game against each other to be the first to mine a block. We extend an existing game-theoretic framework for Bitcoin mining and compute the resultant payoff matrices. From these payoff matrices, we determine optimal quantum mining strategies for two non-colluding and aggressive quantum miners with multiple opportunities at finding a valid block in an otherwise classical Bitcoin network. We show that these optimal quantum mining strategies have a negligible effect on the 51% attack threshold. The novelty of our work is the inclusion of the Aggressive Quantum Mining Strategy and the realistic approach of allowing the quantum miners to restart their search if their measurements do not yield a valid block when determining the optimal quantum mining strategies. Our result is important for evaluating quantum-mining threats on cryptocurrencies based on Proof-of-Work, e.g. Bitcoin

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