  Within the framework of the search for the still unknown exact value of the
real and complex Grothendieck constant $K_G^\mathbb{F}$ in the famous
Grothendieck inequality (unsolved since 1953), where $\mathbb{F}$ denotes
either the real or the complex field, we concentrate our search on their
smallest upper bound. To this end, we establish a basic framework, built on
functions which map correlation matrices to correlation matrices entrywise by
means of the Hadamard product, such as the Krivine function in the real case or
the Haagerup function in the complex case. By making use of multivariate real
and complex Gaussian analysis, higher transcendental functions, integration
over spheres and combinatorics of the inversion of Maclaurin series, we provide
an approach by which we also recover all famous upper bounds of Grothendieck
himself ($K_G^\mathbb{R} \leq \sinh(\pi/2) \approx 2.301$), Krivine
($K_G^\mathbb{R} \leq \frac{\pi}{2 \ln(1 + \sqrt{2})} \approx 1,782$) and
Haagerup ($K_G^\mathbb{C} \leq 1.405$, numerically approximated); each of them
as a special case. In doing so, we aim to unify the real and complex case as
much as possible and apply our results to several concrete examples, including
the Walsh-Hadamard transform (''quantum gate'') and the multivariate Gaussian
copula - with foundations of quantum theory and quantum information theory in
mind. Moreover, we offer a shortening and a simplification of the proof of the
strongest estimation until now; namely that $K_G^\mathbb{R} < \frac{\pi}{2
\ln(1 + \sqrt{2})}$. We summarise our key results in form of an algorithmic
scheme and shed light on related open problems and topics for future research.
