These results imply subexponential lower bounds on the size of proofs needed to refute certain unsatisfiable CNFs in a broad class of proof systems, including tree-like Lovász–Schrijver proofs.Expand

It is found that the proof-of-work used by Bitcoin is relatively resistant to substantial speedup by quantum computers in the next 10 years, mainly because specialized ASIC miners are extremely fast compared to the estimated clock speed of near-term quantum computers.Expand

A stronger version of the adversary method which goes beyond this principle to make explicit use of the stronger condition that the algorithm actually computes the function, and which is a lower bound on bounded-error quantum query complexity.Expand

Two new complexity measures for Boolean functions are introduced, which are named sumPI and maxPI, and the main result is proven via a combinatorial lemma which relates the square of the spectral norm of a matrix to the squares ofthe spectral norms of its submatrices.Expand

Borders are given on the ε-rank of a real matrix A, defined for any ε > 0 as the minimum rank over matrices that approximate every entry of A to within an additive ε.Expand

This work shows an optimal product theorem for discrepancy, namely that for any two Boolean functions f, g, disc(f odot g)=thetas(disc(f) disc(g)).Expand

It is obtained that the general adversary bound characterizes the quantum query complexity of any function whatsoever, implying that discrete and continuous-time query models are equivalent in the bounded-error setting, even for the general state-conversion problem.Expand

These results essentially answer in the affirmative a conjecture of O'Donnell and Servedio that the sign degree--the minimal degree of a polynomial that agrees in sign with a function on the Boolean cube--of every formula of size n is O(sqrt(n).Expand

It is shown that the quantum query complexity of detecting if an n-vertex graph contains a triangle is O(n9/7), and the main theorem shows how this high-level language can be compiled as a learning graph and gives the resulting complexity.Expand

A new technique for proving formula size lower bounds based on matrix rank gives bounds at least as large as those given by the method of Khrapchenko, originally used to prove an n2 lower bound on the parity function.Expand