Microcavity polaritons are hybrid quasiparticles emerging from the strong coupling between quantum well excitons and light in the resonator. Their unique half-light half-matter nature brings in specific properties like low effective mass, nonlinearity due to the repulsive interactions, and ability to directly access the polaritonic wavefunction through optical means. Due to their specific features, polaritons can form macroscopic quantum states and even undergo Bose-Einstein condensation, which makes them very interesting system for studying the quantum phenomena occurring on a macroscopic scale. This thesis is devoted to the investigation of dynamics of the second-order coherence of the microcavity polaritons. The evolution of the coherence of the polariton system occurs on a timescale of few picoseconds only, which required assembly of a new ultrafast photon correlation setup based on the streak-camera. The operation of this setup and requirements for high-quality measurements are discussed in details. After this, the experiments under both nonresonant and resonant excitation of polaritons are reported. Under nonresonant excitation, the formation of the polariton Bose-Einstein condensate is observed, and the change of the polariton coherence during this phase transition is measured. This indicates the time which is required to formthe condensate, and also allows to estimate its propagation velocity. The latter is compared with the measurements of the first-order spatial correlation function. Finally, polaritons confined in two coupled potential traps are studied. The traps are fabricated using the wet etching of the microcavity. In the resulting Josephson junction, polaritons are excited resonantly and demonstrate clear Josephson oscillations between the two traps. At the same time, the correlation measurements demonstrate strong perturbations of the light statistics in phase with the oscillations. This behavior is well represented with the quantum simulations, which also indicate presence of a significant quadrature squeezing in the system. The origin of the squeezing, the perspective for generating the nonclassical light, and other features of this phenomenon are discussed.