Separating the polynomial-time hierarchy by oracles
Proc. 26th annual symposium on Foundations of computer science
Computational limitations of small-depth circuits
Computational limitations of small-depth circuits
Algebraic methods in the theory of lower bounds for Boolean circuit complexity
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
With probability one, a random oracle separates PSPACE from the polynomial-time hierarchy
Journal of Computer and System Sciences - 18th Annual ACM Symposium on Theory of Computing (STOC), May 28-30, 1986
Bounded-width polynomial-size branching programs recognize exactly those languages in NC1
Journal of Computer and System Sciences - 18th Annual ACM Symposium on Theory of Computing (STOC), May 28-30, 1986
PP is as hard as the polynomial-time hierarchy
SIAM Journal on Computing
Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs
Journal of Computer and System Sciences
On the computational power of depth 2 circuits with threshold and modulo gates
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
A weight-size trade-off for circuits with MOD m gates
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
A note on the power of majority gates and modular gates
Information Processing Letters
The power of the middle bit of a #P function
Journal of Computer and System Sciences
Complex polynomials and circuit lower bounds for modular counting
Computational Complexity - Special issue on circuit complexity
Computational Complexity - Special issue on circuit complexity
A complex-number fourier technique for lower bounds on the mod-m degree
Computational Complexity
Upper and Lower Bounds for Some Depth-3 Circuit Classes
CCC '97 Proceedings of the 12th Annual IEEE Conference on Computational Complexity
Lower Bounds for Approximations by Low Degree Polynomials Over Z_m
CCC '01 Proceedings of the 16th Annual Conference on Computational Complexity
Guest Column: correlation bounds for polynomials over {0 1}
ACM SIGACT News
On the Power of Small-Depth Computation
Foundations and Trends® in Theoretical Computer Science
Cracks in the defenses: scouting out approaches on circuit lower bounds
CSR'08 Proceedings of the 3rd international conference on Computer science: theory and applications
ACM Transactions on Computation Theory (TOCT)
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We prove exponentially small upper bounds on the correlation between parity and quadratic polynomials mod 3. One corollary of this is that in order to compute parity, circuits consisting of a threshold gate at the top, mod 3 gates in the middle, and AND gates of fan-in two at the inputs must be of size 2Ω(n). This is the first result of this type for general mod 3 subcircuits with ANDs of fan-in greater than 1. This yields an exponential improvement over a long-standing result of Smolensky, answering a question recently posed by Alon and Beigel. The proof uses a novel inductive estimate of the relevant exponential sums introduced by Cai. Green and Thierauf. The exponential sum and correlation bounds presented here are tight.