Strategies for quantum-enabled Bitcoin miners
Source: arXiv:2607.23952 · Published 2026-07-27 · By Zach Manson, Barry C. Sanders
TL;DR
This paper investigates the impact of two adversarial miners equipped with quantum computers on the security threshold of Bitcoin’s Proof-of-Work (PoW) consensus, focusing specifically on how quantum strategies affect the 51% attack threshold. Unlike previous studies that considered simplistic or restrictive quantum adversarial models, the authors extend a game-theoretic framework to incorporate more realistic quantum mining behavior, including multiple opportunities per mining race and the use of the aggressive quantum mining strategy (AQMS) that allows miners to restart their quantum search to boost success probability. By computing payoff matrices and Nash equilibria for the interaction between these two aggressive quantum miners hidden within a classical Bitcoin network, the paper derives their optimal quantum mining strategies and analyzes their impact through simulation and theoretical analysis. The key finding is that even with these aggressive and realistic quantum strategies, the effect on the 51% attack threshold is negligible, suggesting the Bitcoin network’s security assumptions remain robust in the near-term quantum mining scenario.
Key findings
- Optimal quantum mining strategies are computed for two non-colluding quantum miners performing up to four quantum measurements per mining attempt (Fig 4, 5).
- The aggressive quantum mining strategy (AQMS), which allows miners to restart the quantum search on invalid measurements, increases fork rate but does not significantly lower the 51% attack threshold (Fig 9).
- Across multiple scenarios and Grover iteration budgets (kopt), the 51% attack threshold decreases by less than 1% compared to classical mining baselines (Section IV.C).
- The payoff matrices derived show that both peaceful and aggressive quantum strategies converge to Nash equilibria with limited advantage over classical strategies (Section IV.B).
- Simulation of quantum mining races over 1000 days shows the stale (fork) rate rises modestly but remains below critical levels that would jeopardize network security (Section IV.C).
- Allowing multiple quantum measurements per mining attempt (c > 1) improves individual miners’ expected payoffs but does not translate into easier large-scale attacks (Section IV.B).
- Compared to prior quantum mining threat models restricting to one measurement and peaceful mining, inclusion of AQMS and multiple tries yield more realistic but only marginally stronger adversaries (Section V).
Threat model
The threat model consists of two rational, non-colluding miners equipped with quantum computers capable of running Grover’s algorithm with multiple iterations and measurements per mining race. They compete aggressively within a classical Bitcoin network but cannot collude or manipulate network parameters outside their mining strategy choice. The adversaries can adopt aggressive quantum mining (AQMS) strategies that increase fork rates but cannot control the entire network or classical miners’ behavior. They aim to maximize expected mining rewards and potentially lower the 51% attack threshold but operate within Bitcoin’s standard protocol rules.
Methodology — deep read
Threat Model and Assumptions: The authors consider two quantum miners, Alice and Bob, embedded in an otherwise classical Bitcoin network with many classical miners. These quantum miners do not collude and compete to mine blocks first. They have quantum computers capable of Grover search with multiple iterations and measurements within a single mining race. The adversaries may follow peaceful (PQMS) or aggressive quantum mining strategies (AQMS), with the latter allowing restarting the quantum search if no valid block is found in previous attempts. The adversaries are modeled as rational and aggressive but do not perform classical Byzantine behaviors or collusion.
Data and Setup: No empirical dataset is used; instead, the authors construct mathematical payoff matrices from the quantum mining model parameters. Parameters include the number of allowed quantum measurements c (1 to 4), the Grover iteration budget kopt per measurement, and network difficulty D. The mempool and blockchain protocol parameters are classical.
Architecture and Algorithms: The core is a game-theoretic model extended from Lee et al. that incorporates quantum-specific strategies. Each player selects pure quantum strategies corresponding to the number of Grover iterations per measurement. Payoff matrices A and B map these strategies to expected rewards accounting for success probabilities derived from Grover's algorithm and combining stale rates induced by aggressive quantum mining. The Lemke-Howson algorithm is used to compute exact Nash equilibria of the two-player quantum mining race game.
Training and Computation: The payoff matrices are computed analytically using closed-form formulas for Grover success probabilities combined with the Finite-Budget Quantum Search (FQS) model. Multiple iterations of the Lemke-Howson algorithm run on these matrices to find equilibrium strategies. Simulations over thousands of days sample mining attempts employing these equilibria to observe stale block rates and eventual impact on the 51% attack threshold.
Evaluation Protocol: Metrics include the expected payoff for each miner, the stale (fork) rate in the blockchain, and the adjusted 51% attack threshold computed from network difficulty and hash power. Baselines include classical mining without quantum miners and peaceful quantum mining with a single measurement. Ablations include varying the number of measurements and comparing AQMS to PQMS.
Reproducibility: The authors provide mathematical details sufficient to reproduce the models and computations but do not release code or datasets as all data is synthetic and derived from theoretical frameworks. Some implementation details of simulation parameters and random seeds are not explicitly specified.
Example Workflow: For c=2 measurements per race, each miner chooses k1, k2 Grover iterations. The success probabilities p(k) are computed from the FQS model. Payoff matrices A and B are constructed for all pairs of strategies. The Lemke-Howson algorithm identifies Nash equilibria. Using these strategy distributions, simulations sample daily mining results to compute stale rates and assess impact on network security.
Technical innovations
- Integration of the Aggressive Quantum Mining Strategy (AQMS), allowing miners to restart quantum searches upon invalid measurements, into a two-player quantum mining game-theoretic model.
- Extension of prior quantum mining models to multiple sequential quantum measurements per mining attempt (c > 1), reflecting more realistic quantum mining behavior.
- Derivation and computation of detailed payoff matrices combining Grover iteration budgets, quantum success probabilities, and blockchain stale rates within a competitive two-player setting.
- Application of the Lemke-Howson algorithm to find exact Nash equilibria for the two-player quantum mining race with multiple measurements and AQMS.
Baselines vs proposed
- Classical mining baseline: 51% attack threshold = 51% hash rate; Proposed quantum mining (AQMS, multiple measurements): 51% attack threshold decreases by less than 1%
- Peaceful Quantum Mining Strategy (PQMS) with one measurement: expected payoff lower than AQMS with multiple measurements by ~3% (Section IV.B)
- Simulation stale rate without quantum miners ~1% vs with two quantum miners using AQMS ~2-3% (Section IV.C)
Limitations
- Analysis limited to two quantum miners; does not address coalitions or larger numbers of quantum adversaries.
- Quantum miners are assumed non-colluding and rational; no adversarial Byzantine or network manipulation considered.
- No experimental evaluation with real quantum hardware or noisy intermediate-scale quantum devices.
- The model assumes static classical network parameters, no dynamic difficulty or mempool adjustments during the simulation.
- Did not explore impact on alternative consensus protocols or future Bitcoin upgrades (e.g., Taproot implications).
Open questions / follow-ons
- How do larger coalitions or multiple quantum miners affect the network compared to the two-player model?
- What is the impact of noisy or resource-limited quantum devices on the practical effectiveness of AQMS?
- Could adaptive difficulty adjustments or protocol changes mitigate the modest increases in fork rates caused by quantum miners?
- How would quantum miners interact with other potential quantum attacks on Bitcoin, like those against signature schemes?
Why it matters for bot defense
This paper provides a rigorous game-theoretic and quantum algorithmic analysis of how emerging quantum mining techniques could affect Bitcoin’s Proof-of-Work security, specifically the 51% attack threshold. For bot-defense and CAPTCHA practitioners, the core insight is that even aggressive quantum mining strategies with multiple measurement attempts and restarts do not significantly lower the computational power needed to conduct major consensus attacks. This bolsters confidence that Proof-of-Work networks retain robustness against near-term quantum adversaries. The methods used here—combining algorithmic success probabilities with equilibrium game theory—can inspire analogous modeling approaches for evaluating quantum-enhanced bot attacks or adversarial strategies in security protocols. Moreover, the paper's treatment of strategic measurement retries and their effect on stale block rates could inform defense mechanisms against quantum-accelerated brute force or enumeration attacks on CAPTCHAs or bot detection puzzles.
Cite
@article{arxiv2607_23952,
title={ Strategies for quantum-enabled Bitcoin miners },
author={ Zach Manson and Barry C. Sanders },
journal={arXiv preprint arXiv:2607.23952},
year={ 2026 },
url={https://arxiv.org/abs/2607.23952}
}