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Quantum computers can sometimes exponentially outperform classical ones, butonly for problems with sufficient structure. While it is well known that queryproblems with full permutation symmetry can have at most polynomial quantumspeedup -- even for partial functions -- it is unclear how far this conditionmust be relaxed to enable exponential speedup. In particular, it is natural toask whether exponential speedup is possible for (partial) graph properties, inwhich the input describes a graph and the output can only depend on itsisomorphism class. We show that the answer to this question depends strongly onthe input model. In the adjacency matrix model, we prove that the bounded-errorrandomized query complexity $R$ of any graph property $\mathcal{P}$ has$R(\mathcal{P}) = O(Q(\mathcal{P})^{6})$, where $Q$ is the bounded-errorquantum query complexity. This negatively resolves an open question ofMontanaro and de Wolf in the adjacency matrix model. More generally, we prove$R(\mathcal{P}) = O(Q(\mathcal{P})^{3l})$ for any $l$-uniform hypergraphproperty $\mathcal{P}$ in the adjacency matrix model. In direct contrast, inthe adjacency list model for bounded-degree graphs, we exhibit a promiseproblem that shows an exponential separation between the randomized and quantumquery complexities.

Can graph properties have exponential quantum speedup?