The class QMA(k), introduced by , 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. First, we give a protocol by which a verifier can be convinced that a 3SAT formula of size n is satisfiable, with constant soundness, given O(sqrt(n)) unentangled quantum witnesses with O(log n) qubits each. Our protocol relies on the existence of very short PCPs. Second, 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. Third, we prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one.
Third, Blier and Tapp @cite_17 recently (and independently of us) gave an interesting @math protocol for an @math -complete problem, namely @math . In this protocol, Arthur verifies that an @math -vertex graph @math is @math -colorable, using two unentangled witnesses with only @math qubits each. There is a crucial caveat, though: if @math is @math -colorable, then Arthur can only detect this with probability @math rather than constant probability. Indeed, if the soundness gap were constant rather than @math , then Blier and Tapp's protocol could presumably be scaled up by an exponential to show @math !
@cite_17
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We study a one-parameter family of discrete dynamical systems called the @math -color firefly cellular automata (FCAs), which were introduced recently by the author. At each discrete time @math , each vertex in a graph has a state in @math , and a special state @math is designated as the blinking' state. At step @math , simultaneously for all vertices, the state of a vertex increments from @math to @math unless @math and at least one of its neighbors is in the state @math . A central question about this system is that on what class of network topologies synchrony is guaranteed to emerge. In a previous work, we have shown that for @math , every @math -coloring on a finite tree synchronizes iff the maximum degree is less than @math , and asked whether this behavior holds for all @math . In this paper, we answer the question positively for @math and negatively for all @math by constructing counterexamples on trees with maximum degree at most @math .