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We revisit the so-called compressed oracle technique, introduced by Zhandry for analyzing quantum algorithms in the quantum random oracle model (QROM). To start off with, we offer a concise exposition of the technique, which easily extends to the parallel-query QROM, where in each query-round the considered algorithm may make several queries to the QROM in parallel. This variant of the QROM allows for a more fine-grained query-complexity analysis. Our main technical contribution is a framework that simplifies the use of (the parallel-query generalization of) the compressed oracle technique for proving query complexity results. With our framework in place, whenever applicable, it is possible to prove quantum query complexity lower bounds by means of purely classical reasoning. More than that, for typical examples the crucial classical observations that give rise to the classical bounds are sufficient to conclude the corresponding quantum bounds. We demonstrate this on a few examples, recovering known results (like the optimality of parallel Grover), but also obtaining new results (like the optimality of parallel BHT collision search). Our main target is the hardness of finding a $q$-chain with fewer than $q$ parallel queries, i.e., a sequence $x_0, x_1,\ldots, x_q$ with $x_i = H(x_{i-1})$ for all $1 \leq i \leq q$. The above problem of finding a hash chain is of fundamental importance in the context of proofs of sequential work. Indeed, as a concrete cryptographic application of our techniques, we prove that the "Simple Proofs of Sequential Work" proposed by Cohen and Pietrzak remains secure against quantum attacks. Such an analysis is not simply a matter of plugging in our new bound; the entire protocol needs to be analyzed in the light of a quantum attack. Thanks to our framework, this can now be done with purely classical reasoning.

On the Compressed-Oracle Technique, and Post-Quantum Security of Proofs of Sequential Work