We study the computation power of the congested clique, a model of distributed computation where n players communicate with each other over a complete network in order to compute some function of their inputs. The number of bits that can be sent on any edge in a round is bounded by a parameter b. We consider two versions of the model: in the first, the players communicate by unicast, allowing them to send a different message on each of their links in one round; in the second, the players communicate by broadcast, sending one message to all their neighbors.It is known that the unicast version of the model is quite powerful; to date, no lower bounds for this model are known. In this paper we provide a partial explanation by showing that the unicast congested clique can simulate powerful classes of bounded-depth circuits, implying that even slightly super-constant lower bounds for the congested clique would give new lower bounds in circuit complexity. Moreover, under a widely-believed conjecture on matrix multiplication, the triangle detection problem, studied in [8], can be solved in O(n ) time for any > 0.The broadcast version of the congested clique is the wellknown multi-party shared-blackboard model of communication complexity (with number-in-hand input). This version is more amenable to lower bounds, and in this paper we show that the subgraph detection problem studied in [8] requires polynomially many rounds for several classes of subgraphs. We also give upper bounds for the subgraph detection problem, and relate the hardness of triangle detection in the broadcast congested clique to the communication complexity of set disjointness in the 3-party number-on-forehead model.

The class QMA (k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Does QMA (k) = QMA (2) for k ≥ 2? Can QMA (k) protocols be amplified to exponentially small error?In this paper, we make progress on all of the above questions.• We give a protocol by which a verifier can be convinced that a 3Sat formula of size m is satisfiable, with constant soundness, given O ( √ m) unentangled quantum witnesses with O (log m) qubits each. Our protocol relies on the existence of very short PCPs.• We show that assuming a weak version of the Additivity Conjecture from quantum information theory, any QMA (2) protocol can be amplified to exponentially small error, and QMA (k) = QMA (2) for all k ≥ 2.• We prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one. The class QMA, or Quantum Merlin-Arthur, consists of all languages that admit a proof protocol in which Merlin sends Arthur a polynomial-size quantum state |ψ , and then Arthur decides whether to accept or reject in quantum polynomial time. This class was introduced by Knill [17], Kitaev [15], and Watrous [29] as a quantum analogue of NP. By now we know a reasonable amount about QMA: for example, it allows amplification of success probabilities, is contained in PP, and has natural complete promise problems. (See Aharonov and Naveh [2] for a survey.) In 2003, Kobayashi, Matsumoto, and Yamakami [19] defined a generalization of QMA called QMA (k). Here there are k Merlins, who send Arthur k quantum proofs |ψ 1 , . . . , |ψ k respectively that are guaranteed to be unentangled with each other. (Thus QMA (1) = QMA.) Notice that in the classical case, this generalization is completely uninteresting: we have MA (k) = MA for all k, since we can always simulate k Merlins by a single Merlin who sends Arthur a concatenation of the k proofs. In the quantum case, however, a single Merlin could cheat by entangling the k proofs, and we know of no general way to detect such entanglement.When we try to understand QMA (k), we encounter at least three basic questions. First, do multiple quantum proofs ever actually help? That is, can we find some sort of evidence that QMA (k) = QMA (1) for some k? Second, can QMA (k) protocols be amplified to exponentially small error? Third, are two Merlins the most we ever need? That is, does QMA (k) = QMA (2) for all k ≥ 2? 1 We know of three previous results that are relevant to the above questions. First, in their original paper on QMA (k), Kobayashi et al. [19] proved that a positive answer to the second question implies a positive answer to the third. That is, if QMA (k) protocols can be amplified, then QMA (k) = QMA (2) for all k ≥ 2.Second, Liu, Christandl, and Verstraete [21] gave a natural problem from quantum chemistry, called pure state N -representability, which is in QMA (2) but...

Given an instance of a hard decision problem, a limited goal is to compress that instance into a smaller, equivalent instance of a second problem. As one example, consider the problem where, given Boolean formulas ψ 1 ,. .. , ψ t , we must determine if at least one ψ j is satisfiable. An OR-compression scheme for SAT is a polynomial-time reduction R that maps (ψ 1 ,. .. , ψ t) to a string z, such that z lies in some "target" language L ′ if and only if j [ψ j ∈ SAT] holds. (Here, L ′ can be arbitrarily complex.) AND-compression schemes are defined similarly. A compression scheme is strong if |z| is polynomially bounded in n = max j |ψ j |, independent of t. Strong compression for SAT seems unlikely. Work of Harnik and Naor (FOCS '06/SICOMP '10) and Bodlaender, Downey, Fellows, and Hermelin (ICALP '08/JCSS '09) showed that the infeasibility of strong OR-compression for SAT would show limits to instance compression for a large number of natural problems. Bodlaender et al. also showed that the infeasibility of strong AND-compression for SAT would have consequences for a different list of problems. Motivated by this, Fortnow and Santhanam (STOC '08/JCSS '11) showed that if SAT is strongly ORcompressible, then NP ⊆ coNP/poly. Finding similar evidence against AND-compression was left as an open question. We provide such evidence: we show that strong AND-or OR-compression for SAT would imply non-uniform, statistical zero-knowledge proofs for SAT-an even stronger and more unlikely consequence than NP ⊆ coNP/poly. (By a different argument, we also show such compression would imply NP ⊆ coAM.) Our methods apply against probabilistic compression schemes of sufficient "quality" with respect to the reliability and compression amount (allowing for tradeoff). This greatly strengthens the evidence given by Fortnow and Santhanam against probabilistic OR-compression for SAT. We also give negative results for the analogous task of quantum instance compression, in which a polynomial-time quantum reduction must output a quantum state that, in an appropriate sense, "preserves the answer" to the input instance. The central idea in our proofs is to exploit the information bottleneck in an AND-compression scheme for a language L in order to fool a cheating prover in a proof system for L. Our key technical tool is a new method to "disguise" information being fed into a compressive mapping; we believe this method may find other applications.

BellQMA protocols are a subclass of multi-prover quantum Merlin-Arthur protocols in which the verifier is restricted to perform nonadaptive, unentangled measurements on the quantum states received from each Merlin. In this paper, we prove that m-clause 3-SAT instances have BellQMA proofs of satisfiability with constant soundness gap, in which Õ( √ m) Merlins each send O(log m) qubits to Arthur. Our result answers a question of Aaronson et al., who gave a protocol with similar parameters that used entangled measurements; the analysis of our protocol is significantly simpler than that of Aaronson et al. Our result also complements recent work of Brandao, Christandl, and Yard, who showed upper bounds on the power of multi-prover quantum proofs with unentangled but adaptive (LOCC) measurements.

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