  Here we revisit the quantum algorithms for obtaining Forrelation [Aaronson et
al, 2015] values to evaluate some of the well-known cryptographically
significant spectra of Boolean functions, namely the Walsh spectrum, the
cross-correlation spectrum and the autocorrelation spectrum. We introduce the
existing 2-fold Forrelation formulation with bent duality based promise
problems as desirable instantiations. Next we concentrate on the $3$-fold
version through two approaches. First, we judiciously set-up some of the
functions in $3$-fold Forrelation, so that given an oracle access, one can
sample from the Walsh Spectrum of $f$. Using this, we obtain improved results
than what we obtain from the Deutsch-Jozsa algorithm, and in turn it has
implications in resiliency checking. Furthermore, we use similar idea to obtain
a technique in estimating the cross-correlation (and thus autocorrelation)
value at any point, improving upon the existing algorithms. Finally, we tweak
the quantum algorithm with superposition of linear functions to obtain a
cross-correlation sampling technique. To the best of our knowledge, this is the
first cross-correlation sampling algorithm with constant query complexity. This
also provides a strategy to check if two functions are uncorrelated of degree
$m$. We further modify this using Dicke states so that the time complexity
reduces, particularly for constant values of $m$.
