We review the current status of high-multiplicity double-virtual QCD corrections to processes relevant for LHC phenomenology. In particular, we discuss the recent full-color calculation of the five-parton process, whose two-loop amplitudes are required to obtain next-to-next-to-leading order predictions for three-jet production at the LHC. We address various aspects of the computation, including color decomposition, renormalization, partial amplitudes, color identities and the construction of the finite remainder. We review the method of numerical unitarity, which is used to generate finite-field samples of the amplitude. We then focus on the analytic reconstruction of the coefficient functions from these numerical samples via Ansatz techniques. A novel algorithm, based on the correlation of codimension-one residues, helps manage the complexity of the calculation. Little-group rescalings of the gluon amplitude, inspired by supersymmetry Ward identities, facilitate the computation of the quark amplitudes. We conclude with an outlook towards upcoming computations with an increased number of scales, leading to larger Ans\"atze and more complicated alphabets.