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In this work we first examine the hardness of solving various search problemsby hybrid quantum-classical strategies, namely, by algorithms that have bothquantum and classical capabilities. We then construct a hybridquantum-classical search algorithm and analyze its success probability.Regarding the former, for search problems that are allowed to have multiplesolutions and in which the input is sampled according to arbitrarydistributions we establish their hybrid quantum-classical query complexities --i.e., given a fixed number of classical and quantum queries, determine what isthe probability of solving the search task. At a technical level, our resultsgeneralize the framework for hybrid quantum-classical search algorithmsproposed by Rosmanis. Namely, for an arbitrary distribution $D$ on Booleanfunctions, the probability an algorithm equipped with $\tau_c$ classical and$\tau_q$ quantum queries succeeds in finding a preimage of $1$ for a functionsampled from $D$ is at most $\nu_D \cdot(2\sqrt{\tau_c} + 2\tau_q + 1)^2$,where $\nu_D$ captures the average (over $D$) fraction of preimages of $1$. Asapplications of our hardness results, we first revisit and generalize thesecurity of the Bitcoin protocol called the Bitcoin backbone, to a settingwhere the adversary has both quantum and classical capabilities, presenting anew hybrid honest majority condition necessary for the protocol to properlyoperate. Secondly, we examine the generic security of hash functions againsthybrid adversaries. Regarding our second contribution, we design a hybridalgorithm which first spends all of its classical queries and in the secondstage runs a ``modified Grover'' where the initial state depends on thedistribution $D$. We show how to analyze its success probability for arbitrarytarget distributions and, importantly, its optimality for the uniform and theBernoulli distribution cases.

Generalized Hybrid Search and Applications to Blockchain and Hash Function Security