We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a boolean function is k-linear (a parity function on k variables), we achieve a lower bound of Ω(k) queries, even for adaptive algorithms with two-sided error, thus confirming a conjecture of Goldreich [25]. The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as k-juntas. For some classes, such as the class of monotone functions and the class of s-sparse GF(2) polynomials, we significantly strengthen the best known bounds.

An XOR function is a function of the form g(x, y) = f(x ⊕ y), for some boolean function f on n bits. We study the quantum and classical communication complexity of XOR functions. In the case of exact protocols, we completely characterise one-way communication complexity for all f. We also show that, when f is monotone, g’s quantum and classical complexities are quadratically related, and that when f is a linear threshold function, g’s quantum complexity is Θ(n). More generally, we make a structural conjecture about the Fourier spectra of boolean functions which, if true, would imply that the quantum and classical exact communication complexities of all XOR functions are asymptotically equivalent. We give two randomised classical protocols for general XOR functions which are efficient for certain functions, and a third protocol for linear threshold functions with high margin. These protocols operate in the symmetric message passing model with shared randomness. 1

In this paper we introduce a new technique for proving streaming lower bounds (and one-way communication lower bounds), by reductions from a problem called the Boolean Hidden Hypermatching problem (BHH). BHH is a problem that we introduce and prove the first lower bound for, but it is a generalization of a well-known problem called the Boolean Hidden Matching, that was used by Gavinsky et al. to prove separations between quantum communication complexity and one-way randomized communication complexity. The hardness of the BHH problem is inherently one-way: it is easy to solve using logarithmic two-way communication, but requires √ n communication if Alice is only allowed to send messages to Bob, and not vice-versa. This one-wayness allows us to prove lower bounds, via reductions, for streaming problems and related communication problems whose hardness is also inherently one-way. By designing reductions from BHH, we prove lower bounds for the streaming complexity of approximating the sorting by reversal distance, for approximately counting the number of cycles in a 2-regular graph, and for other problems. For example, here is one lower bound that we prove, for a cycle-counting problem: Alice gets a perfect matching EA on a set of n nodes, and Bob gets a perfect matching EB on the same set of nodes. The union EA ∪ EB is a collection of cycles, and the goal is to approximate the number of cycles in this collection. We prove that if Alice is allowed to send o ( √ n) bits to Bob (and Bob is not allowed to send anything to Alice), then the number of cycles cannot be approximated to within a factor of 1.999, even using a randomized protocol. We prove that it is not even possible to distinguish the case where all cycles are of length 4, from the case where all cycles are of length 8. This lower bound is “natively ” one-way: With 4 rounds of communication, it is easy to distinguish these two cases. 1

Abstract: We prove that coNP � MA in the number-on-forehead model of multiparty communication complexity for up to k =(1−ε)logn players, where ε> 0 is any constant. Specifically, we construct an explicit function F: ({0,1} n) k →{0,1} with conondeterministic complexity O(logn) and Merlin-Arthur complexity nΩ(1). The problem was open for k � 3. As a corollary, we obtain an explicit separation of NP and coNP for up to k =(1−ε)logn players, complementing an independent result by Beame et al. (2010) who separate these classes nonconstructively for up to k = 2 (1−ε)n players. ACM Classification: F.1.3, F.2.3 AMS Classification: 68Q17, 68Q15 Key words and phrases: multiparty communication complexity, nondeterminism, Merlin-Arthur computations, separations and lower bounds 1

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Boolean functions

The investigations were performed at the Centrum Wiskunde & Informatica (CWI) and were supported by Vici grant 639.023.302 from the Netherlands Organization for Scientific Research (NWO).

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Abstract: Entangled quantum systems can exhibit correlations that cannot be simulated classically. For historical reasons such correlations are called “Bell inequality violations.” We give two new two-player games with Bell inequality violations that are stronger, fully explicit, and arguably simpler than earlier work. The first game is based on the Hidden Matching problem of quantum communication complexity, introduced by Bar-Yossef, Jayram, and Kerenidis. This game can be won with probability 1 by a strategy using a maximally entangled state with local dimension n (e. g., logn EPR-pairs), while we show that the winning probability of any classical strategy differs from 1 2 by at most O(log(n)/√n). The second game is based on the integrality gap for Unique Games by Khot and Vishnoi and the quantum rounding procedure of Kempe, Regev, and Toner. Here n-dimensional