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arXiv:2608.30351

An Optimal Separation Between Certificate Complexity and Approximate Degree

Kaspars Balodis

Published Aug 31, 2026• 0 citations
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Abstract

We prove that certificate complexity can be quartically larger than approximate degree. More precisely, we construct a family of total Boolean functions $G$ with $$ C(G) = \tildeΩ(\tilde{deg}(G)^4), $$ where $C$ denotes certificate complexity and $\tilde{deg}$ denotes $1/3$-approximate degree. This is optimal up to polylogarithmic factors, since every total Boolean function $f$ satisfies $C(f)\le O(\tilde{deg}(f)^4)$ by the classical block-sensitivity bounds of Nisan and Nisan--Szegedy. Thus the result closes the gap between these two measures and improves the previously best known separation $C(f)=\tildeΩ(\tilde{deg}(f)^3)$ by Balodis, Ben-David, Göös, Jain, and Kothari. The construction starts from the partial function they used to quadratically separate $0$-certificate complexity from unambiguous $1$-certificate complexity. It already has the required certificate hardness, but its $0$-certificates are unstructured, which blocks the derivation of a low-degree verifier. We keep its $1$-condition and restrict the $0$-inputs to those certified by a structured family whose validity admits a low-degree approximant, while preserving the quadratic hardness. The partial function with its low-degree verifier is then fed through the cheat-sheet framework to yield the total function $G$ with the claimed separation. The main technical ingredient is an approximate polynomial that verifies the certificate in degree $\tilde{O}(\sqrt n)$. The verifier forms a low-degree count $W$ of the candidate $1$-certificates that remain compatible with the asserted $0$-certificate, and tests whether this count is zero. Crucially, the construction ensures that $W$ never exceeds $\tilde{O}(n)$, instead of the $Θ(n^2)$ candidate pairs it counts bringing the verification down to degree $\tilde{O}(\sqrt n)$.

DOI
10.48550/arxiv.2608.30351

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Randomized query complexity can beat certificate complexity

Shalev Ben-David, Robin Kothari

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We prove that certificate complexity can be quartically larger than approximate degree. More precisely, we construct a family of total Boolean functions GGG with C(G)=Ω~(deg~(G)4),C(G) = \tildeΩ(\tilde{deg}(G)^4),C(G)=Ω~(deg~​(G)4), where CCC denotes certificate complexity and deg~\tilde{deg}deg~​ denotes 1/31/31/3-approximate degree. This is optimal up to polylogarithmic factors, since every total Boolean function fff satisfies C(f)≤O(deg~(f)4)C(f)\le O(\tilde{deg}(f)^4)C(f)≤O(deg~​(f)4) by the classical block-sensitivity bounds of Nisan and Nisan--Szegedy. Thus the result closes the gap between these two measures and improves the previously best known separation C(f)=Ω~(deg~(f)3)C(f)=\tildeΩ(\tilde{deg}(f)^3)C(f)=Ω~(deg~​(f)3) by Balodis, Ben-David, Göös, Jain, and Kothari. The construction starts from the partial function they used to quadratically separate 000-certificate complexity from unambiguous 111-certificate complexity. It already has the required certificate hardness, but its 000-certificates are unstructured, which blocks the derivation of a low-degree verifier. We keep its 111-condition and restrict the 000-inputs to those certified by a structured family whose validity admits a low-degree approximant, while preserving the quadratic hardness. The partial function with its low-degree verifier is then fed through the cheat-sheet framework to yield the total function GGG with the claimed separation. The main technical ingredient is an approximate polynomial that verifies the certificate in degree O~(n)\tilde{O}(\sqrt n)O~(n​). The verifier forms a low-degree count WWW of the candidate 111-certificates that remain compatible with the asserted 000-certificate, and tests whether this count is zero. Crucially, the construction ensures that WWW never exceeds O~(n)\tilde{O}(n)O~(n), instead of the Θ(n2)Θ(n^2)Θ(n2) candidate pairs it counts bringing the verification down to degree O~(n)\tilde{O}(\sqrt n)O~(n​).
A Nearly Optimal Lower Bound on the Approximate Degree of AC0^00
The approximate degree of a Boolean function f ⁣:{−1,1}n→{−1,1}f \colon \{-1, 1\}^n \rightarrow \{-1, 1\}f:{−1,1}n→{−1,1} is the least degree of a real polynomial that approximates fff pointwise to error at most 1/31/31/3. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by constant-depth circuits. Specifically, we show how to transform any Boolean function fff with approximate degree ddd into a function FFF on O(n⋅polylog⁡(n))O(n \cdot \operatorname{polylog}(n))O(n⋅polylog(n)) variables with approximate degree at least D=Ω(n1/3⋅d2/3)D = Ω(n^{1/3} \cdot d^{2/3})D=Ω(n1/3⋅d2/3). In particular, if d=n1−Ω(1)d= n^{1-Ω(1)}d=n1−Ω(1), then DDD is polynomially larger than ddd. Moreover, if fff is computed by a polynomial-size Boolean circuit of constant depth, then so is FFF. By recursively applying our transformation, for any constant δ>0δ> 0δ>0 we exhibit an AC0^00 function of approximate degree Ω(n1−δ)Ω(n^{1-δ})Ω(n1−δ). This improves over the best previous lower bound of Ω(n2/3)Ω(n^{2/3})Ω(n2/3) due to Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of nnn that holds for any function. Our lower bounds also apply to (quasipolynomial-size) DNFs of polylogarithmic width. We describe several applications of these results. We give: * For any constant δ>0δ> 0δ>0, an Ω(n1−δ)Ω(n^{1-δ})Ω(n1−δ) lower bound on the quantum communication complexity of a function in AC0^00. * A Boolean function fff with approximate degree at least C(f)2−o(1)C(f)^{2-o(1)}C(f)2−o(1), where C(f)C(f)C(f) is the certificate complexity of fff. This separation is optimal up to the o(1)o(1)o(1) term in the exponent. * Improved secret sharing schemes with reconstruction procedures in AC0^00.
We study how large the certificate complexity C(f){C}(f)C(f) of a total Boolean function can be relative to its randomized and quantum query complexities. We construct a total Boolean function fff whose randomized query complexity with one-sided error satisfies R1(f)=Θ(C(f)){R}_1(f) = Θ(\sqrt{{C}(f)})R1​(f)=Θ(C(f)​). This separation is optimal even when two-sided error is allowed. From this construction, we obtain another total Boolean function FFF with bounded-error quantum query complexity Q(F)=O~(C(F)1/4){Q}(F) = \widetilde O({C}(F)^{1/4})Q(F)=O(C(F)1/4), attaining the general quartic bound up to polylogarithmic factors. For the same function, both exact and zero-error quantum query complexities are O~(C(F))\widetilde O(\sqrt{{C}(F)})O(C(F)​). We also prove matching lower bounds for these two measures on FFF.
A Nearly Optimal Lower Bound on the Approximate Degree of AC0^00
The approximate degree of a Boolean function f ⁣:{−1,1}n→{−1,1}f \colon \{-1, 1\}^n \rightarrow \{-1, 1\}f:{−1,1}n→{−1,1} is the least degree of a real polynomial that approximates fff pointwise to error at most 1/3. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by constant-depth circuits. Specifically, we show how to transform any Boolean function fff with approximate degree ddd into a function FFF on O(n⋅polylog(n))O(n \cdot \mathrm{polylog}(n))O(n⋅polylog(n)) variables with approximate degree at least D=Ω(n1/3⋅d2/3)D = \Omega(n^{1/3} \cdot d^{2/3})D=Ω(n1/3⋅d2/3). In particular, if d=n1−Ω(1)d= n^{1-\Omega(1)}d=n1−Ω(1), then DDD is polynomially larger than ddd. Moreover, if fff is computed by a polynomial-size Boolean circuit of constant depth, then so is FFF. By recursively applying our transformation, for any constant δ>0\delta > 0δ>0 we exhibit an AC0^00 function of approximate degree Ω(n1−δ)\Omega(n^{1-\delta})Ω(n1−δ). This improves upon the best previous lower bound of Ω(n2/3)\Omega(n^{2/3})Ω(n2/3) due to Aaronson and Shi [ J. ACM, 51 (2004), pp. 595--605] and nearly matches the trivial upper bound of nnn that holds for any function. Our lower bounds also apply to (quasipolynomial-size) disjunctive normal forms of polylogarithmic width. We describe several applications of these results and provide the following: (i) for any constant δ>0\delta > 0δ>0, an Ω(n1−δ)\Omega(n^{1-\delta})Ω(n1−δ) lower bound on the quantum communication complexity of a function in AC0^00; (ii) a Boolean function fff with approximate degree at least C(f)2−o(1)C(f)^{2-o(1)}C(f)2−o(1), where C(f)C(f)C(f) is the certificate complexity of fff; this separation is optimal up to the o(1)o(1)o(1) term in the exponent; (iii) improved secret sharing schemes with reconstruction procedures in AC0^00.
A longstanding open question in Boolean function complexity asks whether approximate degree composes multiplicatively under block composition. Although a general multiplicative upper bound is known, matching lower bounds have previously been established only for restricted classes of functions. We resolve this question for all total Boolean functions by proving the matching lower bound. Together with Sherstov's upper bound, our result shows that, for every pair of total Boolean functions f:{0,1}n→{0,1}f:\{0,1\}^n\to\{0,1\}f:{0,1}n→{0,1} and g:{0,1}m→{0,1}g:\{0,1\}^m\to\{0,1\}g:{0,1}m→{0,1}, deg~(f∘g)=Θ ⁣(deg~(f) deg~(g)), \widetilde{deg}(f\circ g) = Θ\!\left( \widetilde{deg}(f)\,\widetilde{deg}(g) \right), deg​(f∘g)=Θ(deg​(f)deg​(g)), where deg~\widetilde{deg}deg​ denotes constant-error approximate degree. We also consider the analogous question for exact quantum query complexity and give a counterexample showing that multiplicative composition fails in this setting.
The approximate non-deterministic degree of a Boolean function fff, denoted ndegε(f)\mathsf{ndeg}_ε(f)ndegε​(f) (written Nε(f)\mathsf{N}_ε(f)Nε​(f) for brevity), is the minimum degree of a real polynomial ppp such that 0≤∣p(x)∣≤ε0 \le |p(x)| \le ε0≤∣p(x)∣≤ε whenever f(x)=0f(x) = 0f(x)=0, and ∣p(x)∣≥1|p(x)| \ge 1∣p(x)∣≥1 whenever f(x)=1f(x) = 1f(x)=1. Unlike exact non-deterministic degree, which only requires the polynomial to be nonzero on 111-inputs, this measure enforces a uniform gap: the polynomial must stay close to zero on all 000-inputs and bounded away from zero on all 111-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 fff, deg(f)≤O~ ⁣(rdeg⁡(f)3). deg(f) \le \widetilde O\!\left(\operatorname{rdeg}(f)^3\right). deg(f)≤O(rdeg(f)3). In their paper, they explicitly propose a stronger conjecture: that approximate degree is polynomially bounded by Nε(f)\mathsf{N}_ε(f)Nε​(f) and Nε(f‾)\mathsf{N}_ε(\overline{f})Nε​(f​) jointly, i.e., for every total Boolean function fff and every constant 0<ε<10<ε<10<ε<1, deg~(f)≤poly⁡(Nε(f),Nε(f‾)). \widetilde{deg}(f) \le \operatorname{poly}(\mathsf N_ε(f), \mathsf N_ε(\overline f)). 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, kkk-uniform hypergraph properties, and read-kkk Disjunctive Normal Form (DNF) formulas.