  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 first study the
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 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 $m$.
