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In the graph collision problem, the task is to decide whether a known graph on $n$ vertices contains an edge with both endpoints marked, given query access to the input bits specifying the marked vertices. We first refute the conjectured $Ω(n^{2/3})$ query lower bound with a bounded error quantum algorithm using $\widetilde O(n^{92/139})$ queries. We then refine this approach to obtain a $\widetilde O(n^{11/17})$ query algorithm. We also obtain a $\widetilde O((nq)^{11/17})$ query bound for all (partial) Boolean output functions with positive certificate complexity at most 2 and alphabet size $q$, contrasting a known $Ω(n^{2/3})$ lower bound for such problems with large alphabet sizes. We complement these algorithms with an $Ω(\sqrt{n\log n})$ quantum query lower bound for graph collision, exceeding the $O(\sqrt n)$ certificate complexity barrier for the positive adversary method.

We consider the complexity implications of proofs of quantumness and (quantum verifier) oblivious state preparation. We show that both of these objects can coexist with the easiness of nondeterministic computation. That is, we give evidence that proofs of quantumness can exist when $\mathsf{P} = \mathsf{NP}$ and that (quantum verifier) oblivious state preparation can exist when $\mathsf{BQP} = \mathsf{QCMA}$. In particular, we give a classical oracle relative to which (1) $\mathsf{P}^\mathcal{O} = \mathsf{NP}^\mathcal{O}$ (2) $\mathsf{BQP}^\mathcal{O} = \mathsf{PSPACE}^\mathcal{O}$ (3) Proofs of quantumness exist. In particular, there exists a language in $\mathsf{IP}^\mathcal{O} \subseteq \mathsf{BQP}^\mathcal{O}$ which is hard on average against $\mathsf{P}^\mathcal{O}/poly$. We give a unitary oracle such that (1) $\mathsf{BQP}^\mathcal{O} = \mathsf{QCMA}^\mathcal{O}$ (2) Quantum verifier oblivious state preparation exists. These oracle separations directly imply a number of related consequences. For example, our first result implies that proofs of quantumness cannot be used to construct classical or quantum cryptography with a relativizing proof. Our second result implies that quantum computation/classical communication oblivious transfer cannot be used to construct post-quantum one-way functions with a relativizing proof.

Fault-tolerant quantum computers run millions of logical cycles to perform quantum error correction (QEC). As logical operations span entire logical cycles, the logical cycle time (LCT) dictates program runtime. Qubit measurements, critical to extract parity information for QEC, are the slowest operations and account for 43-86% of LCT. Measurement latencies impact fidelity: fast measurements lower our ability to correctly classify quantum states, whereas slow measurements increase the likelihood of decoherence. Hardware providers thus calibrate measurement latencies that maximize readout fidelity per qubit and use them to schedule all measurements during program execution. However, calibrated latencies are sub-optimal for logical performance because logical cycles span multiple QEC rounds, during which data qubits are not measured. This increases LCT and logical error rates (LERs) due to decoherence at the logical level. We propose DynaMite, a lightweight real-time measurement latency selection policy built on two key insights. First, logical cycles must use a measurement latency that is fast enough to reduce both LCT and LER for the QEC code, instead of the calibrated latency that only maximizes readout fidelity. Second, not all logical cycles are equal; many do not fully utilize the error budget of the QEC code, i.e., the number of failed parity checks it can tolerate without compromising LER. These cycles could use even faster measurements but do not, because latencies are chosen statically, whereas error information is only available at runtime. DynaMite exploits the fact that a logical cycle comprises multiple QEC rounds and selects the fastest latency for an upcoming round based on the error budget left and rounds remaining. Our evaluations show that DynaMite reduces LCT by 1.69x on average and up to 2.29x, and reduces LER to 0.46x on average and up to 0.13x of baseline.