The starting point of this research proposal is a recent result by the PI, making progress in a half century old, notoriously open problem. In the mid 1960’s, Tukey and Cooley discovered the Fast Fourier Transform, an algorithm for performing one of the most important linear transformations in science and engineering, the (discrete) Fourier transform, in time complexity O(n log n).In spite of its importance, a super-linear lower bound has been elusive for many years, with only very limited results. Very recently the PI managed to show that, roughly speaking, a faster Fourier transform must result in information loss, in the form of numerical accuracy. The result can be seen as a type of computational uncertainty principle, whereby faster computation increases uncertainty in data. The mathematical argument is established by defining a type of matrix quasi-entropy, generalizing Shannon’s measure of information (entropy) to “quasi-probabilities” (which can be negative, more than 1, or even complex).This result, which is not believed to be tight, does not close the book on Fourier complexity. More importantly, the vision proposed by the PI here reaches far beyond Fourier computation. The computation-information tradeoff underlying the result suggests a novel view of complexity theory as a whole. We can now revisit some classic complexity theoretical problems with a fresh view. Examples of these problems include better understanding of the complexity of polynomial multiplication, integer multiplication, auto-correlation and cross-correlation computation, dimensionality reduction via the Fast Johnson-Linednstrauss Transform (FJLT; also discovered and developed by the PI), large scale linear algebra (linear regression, Principal Component Analysis - PCA, compressed sensing, matrix multiplication) as well as binary functions such as integer multiplication.