  We start a systematic study of quarter-BPS operators in four-dimensional
$\mathcal{N}=4$ Super Yang-Mills with gauge group $\mathrm{SU}(N)$ making use
of recently developed tools in conformal field theory. We adapt the technology
of embedding space tensor structures in four dimensions to the problem of
computing R-symmetry tensor structures and we use the underlying chiral algebra
to obtain the superconformal Ward identities. This allows us to fix the
protected part of the four-point correlators, up to few ambiguities. As
applications, we use the Lorentzian inversion formula to study the leading
order OPE data in the large~$N$ supergravity limit and we make contact with the
OPE limit of the five-point function of half-BPS operators.
