In spite of their different natures, light and matter can be unified under the strong-coupling regime, yielding superpositions of the two, referred to as dressed states or polaritons. After initially being demonstrated in bulk semiconductors(1) and atomic systems(2), strong-coupling phenomena have been recently realized in solid-state optical microcavities(3). Strong coupling is an essential ingredient in the physics spanning from many-body quantum coherence phenomena, such as Bose-Einstein condensation(4) and superfluidity(5), to cavity quantum electrodynamics. Within cavity quantum electrodynamics, the Jaynes-Cummings model(6-8) describes the interaction of a single fermionic two-level system with a single bosonic photon mode. For a photon number larger than one, known as quantum strong coupling, a significant anharmonicity is predicted for the ladder-like spectrum of dressed states. For optical transitions in semiconductor nanostructures, first signatures of the quantum strong coupling were recently reported(9). Here we use advanced coherent nonlinear spectroscopy to explore a strongly coupled exciton-cavity system(10,11). We measure and simulate its four-wave mixing response(12,13), granting direct access to the coherent dynamics of the first and second rungs of the Jaynes-Cummings ladder. The agreement of the rich experimental evidence with the predictions of the Jaynes-Cummings model is proof of the quantum strong-coupling regime in the investigated solid-state system.