  We present several key steps towards the computation of differential Higgs
boson cross sections at N$^3$LO in perturbative QCD. Specifically, we work in
the framework of Higgs-differential cross sections that allows to compute
precise predictions for realistic LHC observables. We demonstrate how to
perform an expansion of the analytic N$^3$LO coefficient functions around the
production threshold of the Higgs boson. Our framework allows us to compute to
arbitrarily high order in the threshold expansion and we explicitly obtain the
first two expansion coefficients in analytic form. Furthermore, we assess the
phenomenological viability of threshold expansions for differential
distributions. In addition, we report on an interesting obstacle for the
computation of N$^3$LO corrections with LHAPDF parton distribution functions
and our solution. We provide files containing the analytic expressions for the
partonic cross sections together with the arXiv submission.
