We study the growth of two-and three-body correlations in an ultracold Bose gas quenched to unitarity. This is encoded in the dynamics of the two-and three-body contacts analyzed in this work. Via a set of relations connecting many-body correlations dynamics with few-body models, signatures of the Efimov effect are mapped out as a function of evolution time at unitarity over a range of atomic densities n. For the thermal resonantly interacting Bose gas, we find that atombunching leads to an enhanced growth of few-body correlations. These atom-bunching effects also highlight the interplay between few-body correlations that occurs before genuine many-body effects enter on Fermi timescales.