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Based on the recent breakthrough of Huang (2019), we show that for any total Boolean function f, deg(f) = O(adeg(f)^2): The degree of f is at most quadratic in the approximate degree of f. This is optimal as witnessed by the OR function. D(f) = O(Q(f)^4): The deterministic query complexity of f is at most quartic in the quantum query complexity of f. This matches the known separation (up to log factors) due to Ambainis, Balodis, Belovs, Lee, Santha, and Smotrovs (2017). We apply these results to resolve the quantum analogue of the Aanderaa–Karp–Rosenberg conjecture. We show that if f is a nontrivial monotone graph property of an n-vertex graph specified by its adjacency matrix, then Q(f)=Ω(n), which is also optimal. We also show that the approximate degree of any read-once formula on n variables is Θ(√n).

We present a distribution 𝓓 over inputs in {± 1} 2 N , such that: (1) There exists a quantum algorithm that makes one (quantum) query to the input, and runs in time O (log N ), that distinguishes between 𝓓 and the uniform distribution with advantage Ω (1/log N ). (2) No Boolean circuit of quasi-polynomial size and constant depth distinguishes between 𝓓 and the uniform distribution with advantage better than polylog(N)/√ N . By well-known reductions, this gives a separation of the classes Promise- BQP and Promise- PH in the black-box model and implies an oracle relative to which BQP is not contained in PH .

We present an explicit pseudorandom generator with seed length O((logn)w+1) for read-once, oblivious, width w branching programs that can read their input bits in any order. This improves upon the work of Impagliazzo, Meka and Zuckerman (FOCS’12) where they required seed length n1/2+o(1). A central ingredient in our work is the following bound that we prove on the Fourier spectrum of branching programs. For any width w read-once, oblivious branching program B:{0,1}n→ {0,1}, any k ∈ {1,…,n}, [complex formula not displayed] This settles a conjecture posed by Reingold, Steinke and Vadhan (RANDOM’13). Our analysis crucially uses a notion of local monotonicity on the edge labeling of the branching program. We carry critical parts of our proof under the assumption of local monotonicity and show how to deduce our results for unrestricted branching programs.

A matrix $M: A \times X \rightarrow \{-1,1\}$ corresponds to the following learning problem: An unknown element $x \in X$ is chosen uniformly at random. A learner tries to learn $x$ from a stream of samples, $(a_1, b_1), (a_2, b_2) \ldots$, where for every $i$, $a_i \in A$ is chosen uniformly at random and $b_i = M(a_i,x)$. Assume that $k,\ell, r$ are such that any submatrix of $M$ of at least $2^{-k} \cdot |A|$ rows and at least $2^{-\ell} \cdot |X|$ columns, has a bias of at most $2^{-r}$. We show that any learning algorithm for the learning problem corresponding to $M$ requires either a memory of size at least $\Omega\left(k \cdot \ell \right)$, or at least $2^{\Omega(r)}$ samples. The result holds even if the learner has an exponentially small success probability (of $2^{-\Omega(r)}$). In particular, this shows that for a large class of learning problems, any learning algorithm requires either a memory of size at least $\Omega\left((\log |X|) \cdot (\log |A|)\right)$ or an exponential number of samples, achieving a tight $\Omega\left((\log |X|) \cdot (\log |A|)\right)$ lower bound on the size of the memory, rather than a bound of $\Omega\left(\min\left\{(\log |X|)^2,(\log |A|)^2\right\}\right)$ obtained in previous works [R17,MM17b]. Moreover, our result implies all previous memory-samples lower bounds, as well as a number of new applications. Our proof builds on [R17] that gave a general technique for proving memory-samples lower bounds.

A de Morgan formula over Boolean variables x1,…,xn is a binary tree whose internal nodes are marked with AND or OR gates and whose leaves are marked with variables or their negation. We define the size of the formula as the number of leaves in it. Proving that some explicit function (in P or NP) requires a large formula is a central open question in computational complexity. While we believe that some explicit functions require exponential formula size, currently the best lower bound for an explicit function is the Ω(n3) lower bound for Andreev's function. A long line of work in quantum query complexity, culminating in the work of Reichardt [SODA, 2011], proved that for any formula of size s, there exists a polynomial of degree at most O(√s) that approximates the formula up to a small point-wise error. This is a classical theorem, arguing about polynomials and formulae, however the only known proof for it involves quantum algorithms. We apply Reichardt result to obtain the following: (1) We show how to trade average-case hardness in exchange for size. More precisely, we show that if a function f cannot be computed correctly on more than 1/2 + 2-k of the inputs by any formula of size at most s, then computing f exactly requires formula size at least Ω(k) · s. As an application, we improve the state of the art formula size lower bounds for explicit functions by a factor of Ω(logn). (2) We prove that the bipartite formula size of the Inner-Product function is Ω(n2). (A bipartite formula on Boolean variables x1,…,xn and y1, …, yn is a binary tree whose internal nodes are marked with AND or OR gates and whose leaves can compute any function of either the x or y variables.) We show that any bipartite formula for the Inner-Product modulo 2 function, namely IP(x,y) = Σi=1n xi yi (mod 2), must be of size Ω(n2), which is tight up to logarithmic factors. To the best of our knowledge, this is the first super-linear lower bound on the bipartite formula complexity of any explicit function.

We define a concept class ℱ to be time-space hard (or memory-samples hard) if any learning algorithm for ℱ requires either a memory of size super-linear in n or a number of samples super-polynomial in n, where n is the length of one sample. A recent work shows that the class of all parity functions is time-space hard [Raz, FOCS'16]. Building on [Raz, FOCS'16], we show that the class of all sparse parities of Hamming weight l is time-space hard, as long as l ≥ ω(logn / loglogn). Consequently, linear-size DNF Formulas, linear-size Decision Trees and logarithmic-size Juntas are all time-space hard. Our result is more general and provides time-space lower bounds for learning any concept class of parity functions. We give applications of our results in the field of bounded-storage cryptography. For example, for every ωlogn) ≤ k ≤ n, we obtain an encryption scheme that requires a private key of length k, and time complexity of n per encryption/decryption of each bit, and is provably and unconditionally secure as long as the attacker uses at most o(nk) memory bits and the scheme is used at most 2o(k) times. Previously, this was known only for k=n [Raz, FOCS'16].

For Boolean functions f:{0,1}n -> {0,1} and g:{0,1}m -> {0,1}, the function composition of f and g denoted by f O g : {0,1}nm -> {0,1} is the value of f on n inputs, each of them is the calculation of g on a distinct set of m Boolean variables. Motivated by previous works that achieved some of the best separations between complexity measures such as sensitivity, block-sensitivity, degree, certificate complexity and decision tree complexity we show that most of these complexity measures behave multiplicatively under composition. We use this multiplicative behavior to establish several applications. First, we give a negative answer for Adam Kalai's question from [MOS04]: "Is it true that every Boolean function f:{0,1}n -> {0,1} with degree as a polynomial over the reals (denoted by deg(f)) at most n/3, has a restriction fixing 2n/3 of its variables under which f becomes a parity function?" This question was motivated by the problem of learning juntas as it suggests a simple algorithm, faster than that of Mossel et al. We give a counterexample for the question using composition of functions strongly related to the Walsh-Hadamard code. In fact, we show that for every constants e,δ>0 there are (infinitely many) Boolean functions f: {0,1}n -> {0,1} such that deg(f) ≤ e ⋅ n and under any restriction fixing less than (1-δ) ⋅ n variables, f does not become a parity function.Second, we show that for composition, the block sensitivity (denoted by bs) property has an unusual behavior - namely that bs(f O g) can be larger than bs (f) ⋅ bs(g). We show that the ratio between these two has a strong connection to the integrality gap of the Set Packing problem. In addition, we obtain the best known separation between block-sensitivity and certificate complexity (denoted by C) giving infinitely many functions f such that C(f) ≥ bs (f){log(26)/log(17) = bs (f)1.149....Last, we present a factor 2 improvement of a result by Nisan and Szegedy [NS94], by showing that for all Boolean functions bs f ≤ deg(f)2.

We study the following problem raised by von zur Gathen and Roche [6]: What is the minimal degree of a nonconstant polynomial f: {0,..., n} ź {0,..., m}? Clearly, when m = n the function f(x) = x has degree 1. We prove that when m = n -- 1 (i.e. the point {n} is not in the range), it must be the case that deg(f) = n -- o(n). This shows an interesting threshold phenomenon. In fact, the same bound on the degree holds even when the image of the polynomial is any (strict) subset of {0,..., n}. Going back to the case m = n, as we noted the function f(x) = x is possible, however, we show that if one excludes all degree 1 polynomials then it must be the case that deg(f) = n -- o(n). Moreover, the same conclusion holds even if m=O(n1.475-∈). In other words, there are no polynomials of intermediate degrees that map {0,...,n} to {0,...,m}. Furthermore, we give a meaningful answer when m is a large polynomial, or even exponential, in n. Roughly, we show that if $$m < (_{\,\,\,d}^{n/c} )$$m