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Here we revisit the quantum algorithms for obtaining Forrelation [Aaronson etal, 2015] values to evaluate some of the well-known cryptographicallysignificant spectra of Boolean functions, namely the Walsh spectrum, thecross-correlation spectrum and the autocorrelation spectrum. We introduce theexisting 2-fold Forrelation formulation with bent duality based promiseproblems as desirable instantiations. Next we concentrate on the $3$-foldversion through two approaches. First, we judiciously set-up some of thefunctions in $3$-fold Forrelation, so that given an oracle access, one cansample from the Walsh Spectrum of $f$. Using this, we obtain improved resultsthan what we obtain from the Deutsch-Jozsa algorithm, and in turn it hasimplications in resiliency checking. Furthermore, we use similar idea to obtaina technique in estimating the cross-correlation (and thus autocorrelation)value at any point, improving upon the existing algorithms. Finally, we tweakthe quantum algorithm with superposition of linear functions to obtain across-correlation sampling technique. To the best of our knowledge, this is thefirst cross-correlation sampling algorithm with constant query complexity. Thisalso provides a strategy to check if two functions are uncorrelated of degree$m$. We further modify this using Dicke states so that the time complexityreduces, particularly for constant values of $m$.

Following Forrelation -- Quantum Algorithms in Exploring Boolean Functions' Spectra