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The precise relationship between post-selected classical and 
post-selected quantum computation is an open problem in complexity 
theory. Post-selection has proven to be a useful tool in uncovering some 
of the differences between quantum and classical theories, in 
foundations and elsewhere. This is no less true in the area of 
computational complexity -- quantum computations augmented with 
post-selection are thought to be vastly more powerful than their 
classical counterparts. However, the precise reasons why this might be 
the case are not well understood, and no rigorous separations between 
the two have been found. In this talk, I will consider the difference in 
computational power of classical and quantum post-selection in the 
computational query complexity setting.

We define post-selected classical query algorithms, and relate them to 
rational approximations of Boolean functions; in particular, by showing 
that the post-selected classical query complexity of a Boolean function 
is equal to the minimal degree of a rational function with nonnegative 
coefficients that approximates it (up to a factor of two). For 
post-selected quantum query algorithms, a similar relationship was shown 
by Mahadev and de Wolf, where the rational approximations are allowed to 
have negative coefficients. Using our characterisation, we find an 
exponentially large separation between post-selected classical query 
complexity and post-selected quantum query complexity, by proving a 
lower bound on the degree of rational approximations to the Majority 
function.

One long standing question on quantum non-locality is the quest for an informational principle explaining quantum correlations. Many candidates for such a principle have been proposed with partial success, but none fully explaining quantum correlation. Most of these principles rely on the analysis of the multipartite conditional probabilities. In this talk I will briefly review a few results we had when, instead of analysing multipartite probabilities, we analysed sequences of results from non-local boxes. First, that non-local deterministic boxes cannot be computable. Second, that pseudorandom inputs allow for a local model explaining non-local correlations. Finally, I will present a simple principle that rules out many non-physical sequences of experiments, and a unified view of local and non-signalling correlations based on sequences.

I will describe some general mathematical structures expected to arise from field theories with boundary conditions in terms of factorization algebras, and outline some results and future directions in the study of boundary chiral algebras for 3d N=4 theories following the work of Costello and Gaiotto.