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We prove a general structural theorem for a wide family of local algorithms, which includes property testers, local decoders, and PCPs of proximity. Namely, we show that the structure of every algorithm that makes q adaptive queries and satisfies a natural robustness condition admits a samplebased algorithm with n 1−1/O(q 2 log 2 q) sample complexity. We also prove that this transformation is nearly optimal, and admits a scheme for constructing privacy-preserving local algorithms.Using the unified view that our structural theorem provides, we obtain the following results.• We strengthen the state-of-the-art lower bound for relaxed locally decodable codes, obtaining an exponential improvement on the dependency in query complexity; this resolves an open problem raised by Gur and Lachish (SODA 2020). • We show that any (constant-query) testable property admits a sample-based tester with sublinear sample complexity; this resolves a problem left open in a work of Fischer, Lachish, and Vasudev (FOCS 2015) by extending their main result to adaptive testers. • We prove that the known separation between proofs of proximity and testers is essentially maximal; this resolves a problem left open by Gur and Rothblum (ECCC 2013, Computational Complexity 2018) regarding sublinear-time delegation of computation.Our techniques strongly rely on relaxed sunflower lemmas and the Hajnal-Szemerédi theorem.

We initiate the study of zero-knowledge proofs for data streams. Streaming interactive proofs (SIPs) are well-studied protocols whereby a space-bounded algorithm with one-pass access to a massive stream of data communicates with a powerful but untrusted prover to verify a computation that requires large space.We define the notion of zero-knowledge in the streaming setting and construct zero-knowledge SIPs for the two main building blocks in the streaming interactive proofs literature: the sumcheck and polynomial evaluation protocols. To the best of our knowledge all known streaming interactive proofs are based on either of these tools, and indeed, this allows us to obtain zeroknowledge SIPs for central streaming problems such as index, frequency moments, and inner product. Our protocols are efficient in terms of time and space, as well as communication: the space complexity is polylog(n) and, after a non-interactive setup that uses a random string of near-linear length, the remaining parameters are n o(1) .En route, we develop a toolkit for designing zero-knowledge data stream protocols, consisting of an algebraic streaming commitment protocol and a temporal commitment protocol. The analysis of our protocols relies on delicate algebraic and information-theoretic arguments and reductions from average-case communication complexity.

We initiate the systematic study of QMA algorithms in the setting of property testing, to which we refer as QMA proofs of proximity (QMAPs). These are quantum query algorithms that receive explicit access to a sublinear-size untrusted proof and are required to accept inputs having a property Π and reject inputs that are ε-far from Π, while only probing a minuscule portion of their input.We investigate the complexity landscape of this model, showing that QMAPs can be exponentially stronger than both classical proofs of proximity and quantum testers. To this end, we extend the methodology of Blais, Brody, and Matulef (Computational Complexity, 2012) to prove quantum property testing lower bounds via reductions from communication complexity. This also resolves a question raised in 2013 by Montanaro and de Wolf (cf. Theory of Computing, 2016).Our algorithmic results include a purpose an algorithmic framework that enables quantum speedups for testing an expressive class of properties, namely, those that are succinctly decomposable. A consequence of this framework is a QMA algorithm to verify the Parity of an n-bit string with O(n2/3) queries and proof length. We also propose a QMA algorithm for testing graph bipartitneness, a property that lies outside of this family, for which there is a quantum speedup.

We prove a general structural theorem for a wide family of local algorithms, which includes property testers, local decoders, and PCPs of proximity. Namely, we show that the structure of every algorithm that makes q adaptive queries and satisfies a natural robustness condition admits a sample-based algorithm with n 1−1/O(q 2 log 2 q) sample complexity, following the definition of Goldreich and Ron (TOCT 2016). We prove that this transformation is nearly optimal. Our theorem also admits a scheme for constructing privacy-preserving local algorithms.Using the unified view that our structural theorem provides, we obtain results regarding various types of local algorithms, including the following.• We strengthen the state-of-the-art lower bound for relaxed locally decodable codes, obtaining an exponential improvement on the dependency in query complexity; this resolves an open problem raised by Gur and Lachish (SODA 2020).• We show that any (constant-query) testable property admits a sample-based tester with sublinear sample complexity; this resolves a problem left open in a work of Fischer, Lachish, and Vasudev (FOCS 2015) by extending their main result to adaptive testers.• We prove that the known separation between proofs of proximity and testers is essentially maximal; this resolves a problem left open by Gur and Rothblum (ECCC 2013, Computational Complexity 2018) regarding sublinear-time delegation of computation.Our techniques strongly rely on relaxed sunflower lemmas and the Hajnal-Szemerédi theorem.