The Forrelation problem, first introduced by Aaronson [Scott Aaronson, 2010] and Aaronson and Ambainis [Scott Aaronson and Andris Ambainis, 2015], is a well studied computational problem in the context of separating quantum and classical computational models. Variants of this problem were used to give tight separations between quantum and classical query complexity [Scott Aaronson and Andris Ambainis, 2015]; the first separation between poly-logarithmic quantum query complexity and bounded-depth circuits of super-polynomial size, a result that also implied an oracle separation of the classes BQP and PH [Ran Raz and Avishay Tal, 2019]; and improved separations between quantum and classical communication complexity [Uma Girish et al., 2021]. In all these separations, the lower bound for the classical model only holds when the advantage of the protocol (over a random guess) is more than ≈ 1/√N, that is, the success probability is larger than ≈ 1/2 + 1/√N. This is unavoidable as ≈ 1/√N is the correlation between two coordinates of an input that is sampled from the Forrelation distribution, and hence there are simple classical protocols that achieve advantage ≈ 1/√N, in all these models.
To achieve separations when the classical protocol has smaller advantage, we study in this work the xor of k independent copies of (a variant of) the Forrelation function (where k≪ N). We prove a very general result that shows that any family of Boolean functions that is closed under restrictions, whose Fourier mass at level 2k is bounded by α^k (that is, the sum of the absolute values of all Fourier coefficients at level 2k is bounded by α^k), cannot compute the xor of k independent copies of the Forrelation function with advantage better than O((α^k)/(N^{k/2})). This is a strengthening of a result of [Eshan Chattopadhyay et al., 2019], that gave a similar statement for k = 1, using the technique of [Ran Raz and Avishay Tal, 2019]. We give several applications of our result. In particular, we obtain the following separations:
Quantum versus Classical Communication Complexity. We give the first example of a partial Boolean function that can be computed by a simultaneous-message quantum protocol with communication complexity polylog(N) (where Alice and Bob also share polylog(N) EPR pairs), and such that, any classical randomized protocol of communication complexity at most o(N^{1/4}), with any number of rounds, has quasipolynomially small advantage over a random guess. Previously, only separations where the classical protocol has polynomially small advantage were known between these models [Dmitry Gavinsky, 2016; Uma Girish et al., 2021].
Quantum Query Complexity versus Bounded Depth Circuits. We give the first example of a partial Boolean function that has a quantum query algorithm with query complexity polylog(N), and such that, any constant-depth circuit of quasipolynomial size has quasipolynomially small advantage over a random guess. Previously, only separations where the constant-depth circuit has polynomially small advantage were known [Ran Raz and Avishay Tal, 2019].
The Forrelation problem, first introduced by Aaronson [Scott Aaronson, 2010] and Aaronson and Ambainis [Scott Aaronson and Andris Ambainis, 2015], is a well studied computational problem in the context of separating quantum and classical computational models. Variants of this problem were used to give tight separations between quantum and classical query complexity [Scott Aaronson and Andris Ambainis, 2015]; the first separation between poly-logarithmic quantum query complexity and bounded-depth circuits of super-polynomial size, a result that also implied an oracle separation of the classes BQP and PH [Ran Raz and Avishay Tal, 2019]; and improved separations between quantum and classical communication complexity [Uma Girish et al., 2021]. In all these separations, the lower bound for the classical model only holds when the advantage of the protocol (over a random guess) is more than ≈ 1/√N, that is, the success probability is larger than ≈ 1/2 + 1/√N. This is unavoidable as ≈ 1/√N is the correlation between two coordinates of an input that is sampled from the Forrelation distribution, and hence there are simple classical protocols that achieve advantage ≈ 1/√N, in all these models.
To achieve separations when the classical protocol has smaller advantage, we study in this work the xor of k independent copies of (a variant of) the Forrelation function (where k≪ N). We prove a very general result that shows that any family of Boolean functions that is closed under restrictions, whose Fourier mass at level 2k is bounded by α^k (that is, the sum of the absolute values of all Fourier coefficients at level 2k is bounded by α^k), cannot compute the xor of k independent copies of the Forrelation function with advantage better than O((α^k)/(N^{k/2})). This is a strengthening of a result of [Eshan Chattopadhyay et al., 2019], that gave a similar statement for k = 1, using the technique of [Ran Raz and Avishay Tal, 2019]. We give several applications of our result. In particular, we obtain the following separations:
Quantum versus Classical Communication Complexity. We give the first example of a partial Boolean function that can be computed by a simultaneous-message quantum protocol with communication complexity polylog(N) (where Alice and Bob also share polylog(N) EPR pairs), and such that, any classical randomized protocol of communication complexity at most o(N^{1/4}), with any number of rounds, has quasipolynomially small advantage over a random guess. Previously, only separations where the classical protocol has polynomially small advantage were known between these models [Dmitry Gavinsky, 2016; Uma Girish et al., 2021].
Quantum Query Complexity versus Bounded Depth Circuits. We give the first example of a partial Boolean function that has a quantum query algorithm with query complexity polylog(N), and such that, any constant-depth circuit of quasipolynomial size has quasipolynomially small advantage over a random guess. Previously, only separations where the constant-depth circuit has polynomially small advantage were known [Ran Raz and Avishay Tal, 2019].
The Forrelation problem, first introduced by Aaronson [Scott Aaronson, 2010] and Aaronson and Ambainis [Scott Aaronson and Andris Ambainis, 2015], is a well studied computational problem in the context of separating quantum and classical computational models. Variants of this problem were used to give tight separations between quantum and classical query complexity [Scott Aaronson and Andris Ambainis, 2015]; the first separation between poly-logarithmic quantum query complexity and bounded-depth circuits of super-polynomial size, a result that also implied an oracle separation of the classes BQP and PH [Ran Raz and Avishay Tal, 2019]; and improved separations between quantum and classical communication complexity [Uma Girish et al., 2021]. In all these separations, the lower bound for the classical model only holds when the advantage of the protocol (over a random guess) is more than ≈ 1/√N, that is, the success probability is larger than ≈ 1/2 + 1/√N. This is unavoidable as ≈ 1/√N is the correlation between two coordinates of an input that is sampled from the Forrelation distribution, and hence there are simple classical protocols that achieve advantage ≈ 1/√N, in all these models. To achieve separations when the classical protocol has smaller advantage, we study in this work the xor of k independent copies of (a variant of) the Forrelation function (where k≪ N). We prove a very general result that shows that any family of Boolean functions that is closed under restrictions, whose Fourier mass at level 2k is bounded by α^k (that is, the sum of the absolute values of all Fourier coefficients at level 2k is bounded by α^k), cannot compute the xor of k independent copies of the Forrelation function with advantage better than O((α^k)/(N^{k/2})). This is a strengthening of a result of [Eshan Chattopadhyay et al., 2019], that gave a similar statement for k = 1, using the technique of [Ran Raz and Avishay Tal, 2019]. We give several applications of our result. In particular, we obtain the following separations: Quantum versus Classical Communication Complexity. We give the first example of a partial Boolean function that can be computed by a simultaneous-message quantum protocol with communication complexity polylog(N) (where Alice and Bob also share polylog(N) EPR pairs), and such that, any classical randomized protocol of communication complexity at most õ(N^{1/4}), with any number of rounds, has quasipolynomially small advantage over a random guess. Previously, only separations where the classical protocol has polynomially small advantage were known between these models [Dmitry Gavinsky, 2016; Uma Girish et al., 2021]. Quantum Query Complexity versus Bounded Depth Circuits. We give the first example of a partial Boolean function that has a quantum query algorithm with query complexity polylog(N), and such that, any constant-depth circuit of quasipolynomial size has quasipolynomially small advantage over a random guess. Previously, only separations where the constant-depth circuit has polynomially small advantage were known [Ran Raz and Avishay Tal, 2019].
The 2-Forrelation problem provides an optimal separation between classical and quantum query complexity and is also the problem used for separating BQP and PH relative to an oracle. A natural question is therefore to ask what are the minimal quantum resources needed to solve this problem. We show that 2-Forrelation can be solved using Instantaneous Quantum Polynomial-time (IQP) circuits, a restricted model of quantum computation in which all gates commute. Concretely, signed 2-Forrelation can be solved by a classical random choice between two one-query IQP circuits, while the absolute-value variant uses two independent executions of this randomized procedure. This answers a recent open question of Girish (STOC 2026) on the power of commuting quantum computations. For the Raz-Tal distribution, this randomization is unnecessary. We use this to show that there is an oracle O such that IQPO⊆PHO, strengthening the result of Raz and Tal (STOC 2019). It also yields an oracle separation between IQP and DQC1. We prove Fourier growth bounds for multi-query IQP circuits, including bounds in terms of the size of their accepting set. Our results suggest a possible route toward decision-based quantum advantage within the restricted IQP model. The key ingredient is an algebraic identity of the quadratic function Q(x)=∑i<jxixj that allows extracting inner-product phases within an IQP circuit.
The Forrelation problem, introduced by Aaronson [A10] and Aaronson and Ambainis [AA15], is a well studied problem in the context of separating quantum and classical models. Variants of this problem were used to give exponential separations between quantum and classical query complexity [A10, AA15]; quantum query complexity and bounded-depth circuits [RT19]; and quantum and classical communication complexity [GRT19]. In all these separations, the lower bound for the classical model only holds when the advantage of the protocol (over a random guess) is more than ≈1/N, that is, the success probability is larger than ≈1/2+1/N. To achieve separations when the classical protocol has smaller advantage, we study in this work the XOR of k independent copies of the Forrelation function (where k≪N). We prove a very general result that shows that any family of Boolean functions that is closed under restrictions, whose Fourier mass at level 2k is bounded by αk, cannot compute the XOR of k independent copies of the Forrelation function with advantage better than O(Nk/2αk). This is a strengthening of a result of [CHLT19], that gave a similar result for k=1, using the technique of [RT19]. As an application of our result, we give the first example of a partial Boolean function that can be computed by a simultaneous-message quantum protocol of cost $\mbox{polylog}(N)$ (when players share $\mbox{polylog}(N)$ EPR pairs), however, any classical interactive randomized protocol of cost at most o~(N1/4), has quasipolynomially small advantage over a random guess. We also give the first example of a partial Boolean function that has a quantum query algorithm of cost $\mbox{polylog}(N)$, and such that, any constant-depth circuit of quasipolynomial size has quasipolynomially small advantage over a random guess.
The level-k ℓ1-Fourier weight of a Boolean function refers to the sum of absolute values of its level-k Fourier coefficients. Fourier growth refers to the growth of these weights as k grows. It has been extensively studied for various computational models, and bounds on the Fourier growth, even for the first few levels, have proven useful in learning theory, circuit lower bounds, pseudorandomness, and quantum-classical separations.In this work, we investigate the Fourier growth of certain functions that naturally arise from communication protocols for XOR functions (partial functions evaluated on the bitwise XOR of the inputs x and y to Alice and Bob). If a protocol C computes an XOR function, then C(x,y) is a function of the parity x⊕y. This motivates us to analyze the XOR-fiber of the communication protocol C, defined as h(z):=Ex,y[C(x,y)∣x⊕y=z].We present improved Fourier growth bounds for the XOR-fibers of randomized protocols that communicate d bits. For the first level, we show a tight O(d) bound and obtain a new coin theorem, as well as an alternative proof for the tight randomized communication lower bound for the Gap-Hamming problem. For the second level, we show an d3/2⋅polylog(n) bound, which improves the previous O(d2) bound by Girish, Raz, and Tal (ITCS 2021) and implies a polynomial improvement on the randomized communication lower bound for the XOR-lift of the Forrelation problem, which extends the quantum-classical gap for this problem.Our analysis is based on a new way of adaptively partitioning a relatively large set in Gaussian space to control its moments in all directions. We achieve this via martingale arguments and allowing protocols to transmit real values. We also show a connection between Fourier growth and lifting theorems with constant-sized gadgets as a potential approach to prove optimal bounds for the second level and beyond.
We study the randomized classical query complexity of cyclic Forrelation, a cyclic analogue of k-Forrelation [AA15, STOC'15; BS21, STOC'21]. Given k Boolean functions f1,…,fk:F2n→{−1,1}, the goal is to estimate Tk=N−1Tr(Df1H⊗n⋯DfkH⊗n), where Dfi=diag(fi(x):x∈F2n), H is the Hadamard gate, and N=2n. This normalized-trace problem can be solved with k quantum queries in the DQC1 model. Its classical query complexity was left open in [JLS+26, FOCS'26], which conjectured a lower bound of Ω(N1−2/k).
We establish a nearly complete characterization of the nonadaptive classical query complexity of cyclic Forrelation. For even k≥4, we prove a tight bound of Θk(N1−2/k) for constant additive error. For odd k≥5, we obtain a lower bound of Ωk(N1−2/(k−1)) and an upper bound of Ok(N1−2/k). Surprisingly, k=3 admits a polylogarithmic-query nonadaptive classical algorithm, disproving the conjecture in this case. These results suggest an unexpected parity-dependent structure: the nonadaptive complexity may be governed by the independence number of the k-cycle, leading to the exponent 1−1/⌊k/2⌋, rather than the previously conjectured 1−2/k, which was motivated by a Fourier-growth heuristic. Finally, allowing adaptivity, we prove a lower bound of Ω(N1/4) for every k≥4.
Although formally close to k-Forrelation, the cyclic structure leads to rather different classical behavior and requires different techniques. Our lower bounds are based on hidden-shift collision constructions, and our upper bounds follow from randomized sampling together with cycle-specific conditional-moment estimates. Our results provide a systematic understanding of its query advantages over classical computation.
BQP⊆IP Does Not Relativize
We construct an oracle relative to which BQP⊆IP, resolving a long-standing open question in quantum complexity theory. Together with recent work due to Aaronson et al., our work also gives the first oracle separation between IP and MIP, answering a question dating back to Fortnow's thesis. Our separation is based on the Forrelation problem, where given Boolean functions f and g, the goal is to determine if f is correlated with the Fourier spectrum of g. While this task is solvable by a query-efficient quantum algorithm, we show that it admits no classical interactive protocol with polynomial communication and a polynomial-query verifier.
Our proof is based on (i) a new structural result showing how to approximate Avg-Max circuits (which are well-known to capture the power of interactive proofs in the oracular setting) by convex functions with small first and second derivatives and (ii) a novel analysis establishing that the Forrelation distribution suggested by Aaronson and Ambainis fools such functions.
Our results imply that any prover-efficient classical interactive protocol for BQP must rely on non-relativizing techniques. This might serve as a partial explanation for the lack of progress towards doubly-efficient, unconditionally sound classical verification of quantum computation.