Abstract: We show that new tools from the formal study of scattering amplitudes can be applied to two-loop QCD corrections at the LHC, required to reduce the theoretical uncertainties in the cross sections to a few percent. The key lessons are complex momenta and hidden symmetries. The method of numerical unitarity overcomes the worse-than-factorial growth of the number of Feynman diagrams as the number of loops and legs increases, but previously has only been applied to QCD at the one-loop level. We present the first calculation of a phenomenologically interesting two-loop QCD amplitude using numerical unitarity, by exploiting integration-by-parts (IBP) reduction in a unitarity-compatible manner without raised propagator powers. Surprisingly, additional insights come from dual conformal symmetry, most studied as a hidden symmetry of N=4 SYM. In nonsupersymmetric theories like QCD, the remnant of dual conformal symmetry naturally leads to unitarity-compatible IBP relations and differential equations for Feynman integrals. As a side product, we find an intimate connection between IBP reduction and the Landau equations for infrared singularities.