Exact and point-wise approximating representations of Boolean functions by real polynomials have been of great interest in the theory of computing. We focus on the study of sparsity of such representations. Our results include the following: First, we show that for every total Boolean function, its exact and approximate sparsity in the De Morgan basis are polynomially related to each other in the log scale, ignoring poly-log(n) factors. This answers an open question posed by Knop, Lovett, McGuire and Yuan (STOC 2021). It builds on and is analogous to the seminal result of Nisan and Szegedy (Computational Complexity 1994) who proved the same for degree and approximate degree. Second, we consider more powerful representations using generalized monomials, where each monomial is an indicator of a sub-cube. There are 3n such monomials, where n is the number of variables. We prove that even for these representations, the sparsity and approximate sparsity of total Boolean functions remain polynomially related to each other in the log scale, ignoring poly-log(n) factors. Third, we show that for every total Boolean function f, the log of its De Morgan sparsity characterizes up to polynomial loss and ignoring poly-log(n) factors, the quantum and classical 2-party bounded-error communication complexity of f ∘ EQ4, where EQ4 is Equality of two 2-bit strings, one held by Alice and the other by Bob. As a consequence, we show that bounded-error quantum protocols cannot exhibit super-polynomial cost advantage over their classical counterparts, for computing such functions.
Exact and point-wise approximating representations of Boolean functions by real polynomials have been of great interest in the theory of computing. We focus on the study of sparsity of such representations. Our results include the following: First, we show that for every total Boolean function, its exact and approximate sparsity in the De Morgan basis are polynomially related to each other in the log scale, ignoring poly-log(n) factors. This answers an open question posed by Knop, Lovett, McGuire and Yuan (STOC 2021). It builds on and is analogous to the seminal result of Nisan and Szegedy (Computational Complexity 1994) who proved the same for degree and approximate degree. Second, we consider more powerful representations using generalized monomials, where each monomial is an indicator of a sub-cube. There are 3n such monomials, where n is the number of variables. We prove that even for these representations, the sparsity and approximate sparsity of total Boolean functions remain polynomially related to each other in the log scale, ignoring poly-log(n) factors. Third, we show that for every total Boolean function f, the log of its De Morgan sparsity characterizes up to polynomial loss and ignoring poly-log(n) factors, the quantum and classical 2-party bounded-error communication complexity of f ∘ EQ4, where EQ4 is Equality of two 2-bit strings, one held by Alice and the other by Bob. As a consequence, we show that bounded-error quantum protocols cannot exhibit super-polynomial cost advantage over their classical counterparts, for computing such functions.
A major open problem in quantum communication complexity is whether quantum protocols can be exponentially more efficient than classical protocols for computing total Boolean functions; the prevailing conjecture is that they cannot be so. In a seminal work, Razborov (2002) resolved this question for AND-functions of the form F(x,y)=f(x1∧y1,…,xn∧yn), when the outer function f is symmetric, by proving that their bounded-error quantum and classical communication complexities are polynomially related. Since then, extending this result to all AND-functions has remained open and has been posed by several authors.
In this work, we settle this problem in a strong way. We show that for every Boolean function f, the bounded-error quantum and classical deterministic communication complexities of the function f∘AND2 are polynomially related, up to polylogarithmic factors in n. We prove this by showing that both are characterized--up to polynomial loss--by the logarithm of the De Morgan sparsity of f.
Our results build on the recent work of Chattopadhyay, Dahiya, and Lovett (2025) on structural characterizations of non-sparse Boolean functions, which we extend to resolve the conjecture for general AND-functions.
A major open problem in quantum communication complexity is whether quantum protocols can be exponentially more efficient than classical protocols for computing total Boolean functions; the prevailing conjecture is that they cannot be so. In a seminal work, Razborov (2002) resolved this question for And-functions of the form F(x,y) = f(x₁ ∧ y₁, …, x_n ∧ y_n), when the outer function f is symmetric, by proving that their bounded-error quantum and classical communication complexities are polynomially related. Since then, extending this result to all And-functions has remained open and has been posed by several authors. In this work, we settle this problem in a strong way. We show that for every Boolean function f, the bounded-error quantum and classical deterministic communication complexities of the function f∘And₂ are polynomially related, up to polylogarithmic factors in n. We prove this by showing that both are characterized - up to polynomial loss - by the logarithm of the De Morgan sparsity of f. Our results build on the recent work of Chattopadhyay, Dahiya, and Lovett [Arkadev Chattopadhyay et al., 2026] on structural characterizations of non-sparse Boolean functions, which we extend to resolve the conjecture for general And-functions.
Nisan and Szegedy (CC 1994) showed that any Boolean function f : {0, 1}n → {0, 1} that depends on all its input variables, when represented as a real-valued multivariate polynomial P(x1,..., xn), has degree at least log n − O(log log n). This was improved to a tight (log n − O(1)) bound by Chiarelli, Hatami and Saks (Combinatorica 2020). Similar statements are also known for other Boolean function complexity measures such as Sensitivity (Simon (FCT 1983)), Quantum query complexity, and Approximate degree (Ambainis and de Wolf (CC 2014)). In this paper, we address this question for Probabilistic degree. The function f has probabilistic degree at most d if there is a random real-valued polynomial of degree at most d that agrees with f at each input with high probability. Our understanding of this complexity measure is significantly weaker than those above: for instance, we do not even know the probabilistic degree of the OR function, the best-known bounds put it between (log n)1/2−o(1) and O(log n) (Beigel, Reingold, Spielman (STOC 1991); Tarui (TCS 1993); Harsha, Srinivasan (RSA 2019)). Here we can give a near-optimal understanding of the probabilistic degree of n-variate functions f, modulo our lack of understanding of the probabilistic degree of OR. We show that if the probabilistic degree of OR is (log n)c, then the minimum possible probabilistic degree of such an f is at least (log n)c/(c+1)−o(1), and we show this is tight up to (log n)o(1) factors.
The approximate non-deterministic degree of a Boolean function f, denoted ndegε(f) (written Nε(f) for brevity), is the minimum degree of a real polynomial p such that 0≤∣p(x)∣≤ε whenever f(x)=0, and ∣p(x)∣≥1 whenever f(x)=1. Unlike exact non-deterministic degree, which only requires the polynomial to be nonzero on 1-inputs, this measure enforces a uniform gap: the polynomial must stay close to zero on all 0-inputs and bounded away from zero on all 1-inputs.
The rational degree conjecture, open for over three decades, was recently resolved by Kothari, Kovacs-Deak, Wang, and Yang, who showed that for every total Boolean function f, deg(f)≤O(rdeg(f)3).
In their paper, they explicitly propose a stronger conjecture: that approximate degree is polynomially bounded by Nε(f) and Nε(f) jointly, i.e., for every total Boolean function f and every constant 0<ε<1, deg(f)≤poly(Nε(f),Nε(f)).
This conjecture, if true, would imply a polynomial version of the rational degree result and bring us closer to resolving de Wolf's longstanding non-deterministic degree conjecture.
In this work, we make the first systematic progress on this problem, establishing the conjecture for several broad and natural function classes: monotone and unate functions, functions of bounded alternation number, symmetric functions, k-uniform hypergraph properties, and read-k Disjunctive Normal Form (DNF) formulas.
We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on n input bits, each of which has approximate Fourier sparsity at most O(n³) and randomized parity decision tree complexity Θ(n). This improves upon the recent work of Chattopadhyay, Mande and Sherif [Chattopadhyay et al., 2020] both qualitatively (in terms of designing a large number of examples) and quantitatively (shrinking the gap from quartic to cubic). We leave open the problem of proving a randomized communication complexity lower bound for XOR compositions of our examples. A linear lower bound would lead to new and improved refutations of the log-approximate-rank conjecture. Moreover, if any of these compositions had even a sub-linear cost randomized communication protocol, it would demonstrate that randomized parity decision tree complexity does not lift to randomized communication complexity in general (with the XOR gadget).
Nisan and Szegedy (CC 1994) showed that any Boolean function f:{0,1}n→{0,1} that depends on all its input variables, when represented as a real-valued multivariate polynomial P(x1,…,xn), has degree at least logn−O(loglogn). This was improved to a tight (logn−O(1)) bound by Chiarelli, Hatami and Saks (Combinatorica 2020). Similar statements are also known for other Boolean function complexity measures such as Sensitivity (Simon (FCT 1983)), Quantum query complexity, and Approximate degree (Ambainis and de Wolf (CC 2014)). In this paper, we address this question for \emph{Probabilistic degree}. The function f has probabilistic degree at most d if there is a random real-valued polynomial of degree at most d that agrees with f at each input with high probability. Our understanding of this complexity measure is significantly weaker than those above: for instance, we do not even know the probabilistic degree of the OR function, the best-known bounds put it between (logn)1/2−o(1) and O(logn) (Beigel, Reingold, Spielman (STOC 1991); Tarui (TCS 1993); Harsha, Srinivasan (RSA 2019)). Here we can give a near-optimal understanding of the probabilistic degree of n-variate functions f, \emph{modulo} our lack of understanding of the probabilistic degree of OR. We show that if the probabilistic degree of OR is (logn)c, then the minimum possible probabilistic degree of such an f is at least (logn)c/(c+1)−o(1), and we show this is tight up to (logn)o(1) factors.