This paper gives quantum query lower bounds for two relational graph problems, triangle listing and explicit multiplicative spanner construction, in the general graph query model, where quantum adjacency, degree and neighborhood queries are all available in arbitrary superposition. Both results are obtained by reductions via intermediate multi-block search problems. We extend the quantum query recording framework by Zhandry (CRYPTO 2019) and Hamoudi and Magniez (ToCT 2023) to handle a recording architecture for bidirectional oracles and give a generic blockwise soundness framework, which for arbitrary families of local accepting projectors, gives an exact operator-norm characterization of their maximum overlap with the subspace of bounded weight records. Using the intermediate search problems, we exhibit a family of $n$-vertex graphs with $Θ(n)$ triangles on which listing any constant fraction of the triangles requires $Ω(n^{3/2-o(1)})$ quantum queries. This is the first nontrivial quantum lower bound for triangle listing, a question raised by Jiang and Peng (ICML 2026). Further, we show that, for every fixed $k\ge 7$, constructing a multiplicative $k$-spanner requires $Ω(n^{1+\frac{1}{2μ_k}})$ quantum queries, where $μ_k=k/3+O(1)$. For $k\in\{7,8\}$ the bound is $Ω(n^{5/4})$, which matches what would be implied by an unproven instance of the Erdős girth conjecture, and for large $k$ the exponent $1+\frac{3}{2k}$ exceeds the $1+\frac{4}{3k}$ implied by the provable high girth dense graphs due to Lazebnik, Ustimenko and Woldar, 1995.
This paper gives quantum query lower bounds for two relational graph problems, triangle listing and explicit multiplicative spanner construction, in the general graph query model, where quantum adjacency, degree and neighborhood queries are all available in arbitrary superposition. Both results are obtained by reductions via intermediate multi-block search problems. We extend the quantum query recording framework by Zhandry (CRYPTO 2019) and Hamoudi and Magniez (ToCT 2023) to handle a recording architecture for bidirectional oracles and give a generic blockwise soundness framework, which for arbitrary families of local accepting projectors, gives an exact operator-norm characterization of their maximum overlap with the subspace of bounded weight records. Using the intermediate search problems, we exhibit a family of n-vertex graphs with Θ(n) triangles on which listing any constant fraction of the triangles requires Ω(n3/2−o(1)) quantum queries. This is the first nontrivial quantum lower bound for triangle listing, a question raised by Jiang and Peng (ICML 2026). Further, we show that, for every fixed k≥7, constructing a multiplicative k-spanner requires Ω(n1+2μk1) quantum queries, where μk=k/3+O(1). For k∈{7,8} the bound is Ω(n5/4), which matches what would be implied by an unproven instance of the Erdős girth conjecture, and for large k the exponent 1+2k3 exceeds the 1+3k4 implied by the provable high girth dense graphs due to Lazebnik, Ustimenko and Woldar, 1995.
We consider the quantum query complexity of local search as a function of graph geometry. Given a graph G=(V,E) with n vertices and black box access to a function f:V→R, the goal is find a vertex v that is a local minimum, i.e. with f(v)≤f(u) for all (u,v)∈E, using as few oracle queries as possible. We show that the quantum query complexity of local search on G is Ω(gn43), where g is the vertex congestion of the graph. For a β-expander with maximum degree Δ, this implies a lower bound of $ Ω\bigl(\frac{\sqrtβ \; n^{\frac{1}{4}}}{\sqrtΔ \; \log{n}} \bigr)$. We obtain these bounds by applying the strong weighted adversary method to a construction by Brânzei, Choo, and Recker (2024). As a corollary, on constant degree expanders, we derive a lower bound of Ω(lognn41). This improves upon the best prior quantum lower bound of $Ω\bigl( \frac{n^{\frac{1}{8}}}{\log{n}}\bigr) $ by Santha and Szegedy (2004). In contrast to the classical setting, a gap remains in the quantum case between our lower bound and the best-known upper bound of O(n31) for such graphs.
We study the quantum query complexity of several basic graph problems. 1) We prove a tight Θ(n3/2) bound for finding a maximum-out-degree vertex in an n-vertex tournament, resolving a conjecture of Mande, Paraashar, and Saurabh [FSTTCS'23]. 2) Using the same proof idea, we give a direct proof that finding a maximum-degree vertex in an n-vertex undirected graph requires Θ(n3/2) quantum queries, a bound implicit in the work of Apers and Lee [CCC'21]. Both lower bounds above hold even for computing only the maximum degree or maximum out-degree. 3) We also study Maximum (k,n−k)-Cut, which asks for the largest cut with one side of size k. For every constant k≥1, we give an O(n2−1/(k+1))-query quantum algorithm. 4) We establish a reduction to Maximum (k,n−k)-Cut involving k-Graph Collision, which asks whether the marked vertices of a known graph contain a k-clique. For k≥3, this yields unconditional lower bounds with exponents strictly larger than 3/2, showing that this problem is strictly harder than computing maximum degree. Additionally, under a conjecture we propose for k-Graph Collision, the reduction yields a lower bound matching our upper bound above.
We study the minimum weight cycle problem (MWC) and the minimum weight triangle problem (MWT) in the quantum query model, for n-vertex graphs with integer edge weights in [M] given by their adjacency matrix. Roditty and Vassilevska Williams showed that the two problems are classically equivalent. We show that an exact minimum weight cycle of an undirected graph can be found with O~(n19/12) queries, while Ω(n3/2) queries are necessary. The algorithm learns only the n7/6 lightest vertex pairs, uses quantum counting to locate weight thresholds above them, applies a strict form of the Moore bound to bound the optimum and the size of each weight class, and searches the classes for the few heavy edges of an optimal cycle with nested quantum walks. A simpler version gives O~(n5/3) queries, and searching for fewer heavy edges trades queries for approximation, with a 2-approximation in O~(n8/5) queries and a tight O(logn)-approximation in O~(n3/2) queries. In contrast, MWT costs O(n9/7logM) queries, so the classical equivalence becomes a polynomial separation. For cycles of constant length we give explicit exponents, such as O~(n125/83) for the minimum weight 5-cycle, and we prove that deciding whether a graph has a k-cycle of total weight exactly zero requires Ω(n2−2/(⌊k/2⌋+1)) queries for every constant k≥4. Finally, non-adaptive learning graphs cannot beat n2 for cycles of about logn edges, and we conjecture that exact MWC on directed graphs requires n2−o(1) queries.
Subgraph detection asks whether an n-vertex graph, accessed through queries to its adjacency matrix, contains a copy of a fixed graph H. We prove the first unconditional superlinear lower bounds on the bounded-error quantum query complexity of this problem, answering a longstanding open question. A copy of H is a certificate of constant size, so the adversary method with nonnegative weights cannot prove superlinear lower bounds.
For every fixed r≥4, detecting the clique Kr requires nλr−o(1) queries, where λ4=19/18, the exponents λr increase strictly with r, and λr≥2−42/r+O(1/r). More generally, we prove superlinear lower bounds for detecting every fixed connected graph H with chromatic number c≥4. These bounds approach quadratic as c grows: for sufficiently large c, detection requires n2−O(loglogc/c)−o(1) queries. Chromatic number alone does not characterize the quantum query complexity of subgraph detection: we show that detecting the complete bipartite graph Kr,r requires nβr−o(1) queries, where β10=181/180 and βr≥2−O(1/r).
Our main technical result is a lower bound for finding an all-ones certificate from a known family when the input bits are sampled independently. Its proof combines Zhandry's compressed oracle [CRYPTO 2019] with conditioning on a randomly planted certificate, adapting an argument of Belovs [FOCS 2026]. Our hard instances are built from graphs containing many copies of the desired subgraph with limited overlap. For cliques, we use a construction of Gowers and Janzer [CPC 2021]; for complete bipartite graphs, we use a random construction.
The edge list model is arguably the simplest input model for graphs, where the graph is specified by a list of its edges. In this model, we study the quantum query complexity of three variants of the triangle finding problem. The first asks whether there exists a triangle containing a target edge and raises general questions about the hiding of a problem's input among irrelevant data. The second asks whether there exists a triangle containing a target vertex and raises general questions about the shuffling of a problem's input. The third asks whether there exists a triangle; this problem bridges the 3-distinctness and 3-sum problems, which have been extensively studied by both cryptographers and complexity theorists. We provide tight or nearly tight results for these problems as well as some first answers to the general questions they raise. Furthermore, given any graph with low maximum degree, such as a typical random sparse graph, we prove that the quantum query complexity of finding a length-k cycle in its length-m edge list is m3/4−1/(2k+2−4)±o(1), which matches the best-known upper bound for the quantum query complexity of k-distinctness on length-m inputs up to an mo(1) factor. We prove the lower bound by developing new techniques within Zhandry's recording query framework [CRYPTO '19] as generalized by Hamoudi and Magniez [ToCT '23]. These techniques extend the framework to treat any non-product distribution that results from conditioning a product distribution on the absence of rare events. We prove the upper bound by adapting Belovs's learning graph algorithm for k-distinctness [FOCS '12]. Finally, assuming a plausible conjecture concerning only cycle finding, we show that the lower bound can be lifted to an essentially tight lower bound on the quantum query complexity of k-distinctness, which is a long-standing open question.
The constructive diagonal Ramsey problem asks, given adjacency-oracle access to an N-vertex graph, for a clique or independent set of the order guaranteed by Ramsey's theorem. We give a bounded-error quantum algorithm that, for every K≥2 and N≥4K−1, finds and verifies a homogeneous K-set using O(2KKlogηK) edge queries with failure probability at most η. At the Ramsey scale N=2n, this yields a homogeneous set of order ⌊n/2⌋+1 using O(NlogNlog(logN/η)) queries, improving on the O(N) queries of the explicit classical recursion and giving, to our knowledge, the first sublinear worst-case algorithm for the Ramsey relation. We also derive an Ω(N1/12) quantum lower bound by a reduction from collision finding.
The algorithm runs the constructive recursion over implicit candidate sets. Each set is represented by a short conjunction of adjacency constraints and sampled using capped unknown-solution quantum search, and a scale-aware concentration schedule balances estimation accuracy against the increasing cost of sampling deeper sets. We complement the upper bound with an Ω(N1−1/2) randomized lower bound, transported from the random-Painter analysis of online Ramsey numbers, which holds on the uniform distribution G(N,1/2). On that distribution a greedy quantum search uses only O(N1/4) queries, giving a provable polynomial quantum speedup for Ramsey search on random graphs. We also give an estimation-free size-biased recursion and extend it to every fixed number of edge colours.