With the detection of gravitational waves from merging binary black holes, over the last few years, there has been a considerable interest in the literature to understand if these black holes could have originated in the early universe. If the primordial scalar power over small scales is boosted considerably when compared to the COBE normalized amplitude over large scales, then, such an increased power can lead to a copious production of primordial black holes that can constitute a significant fraction of the cold dark matter density today. Recently, many models of inflation involving single or two scalar fields have been constructed which lead to enhanced power on small scales. In this work, we examine the evolution of the quantum state of the curvature perturbations in such single and two field models of inflation using measures of squeezing, entanglement entropy or quantum discord. We find that, in the single as well as the two field models, the extent of squeezing of the modes is enhanced to a certain extent (when compared to the scenarios involving only slow roll) over modes which exhibit increased scalar power. Moreover, we show that, in the two field models, the entanglement entropy associated with the curvature perturbation, arrived at when the isocurvature perturbation has been traced out, is also enhanced over the small scales. We also take the opportunity to discuss the relation between entanglement entropy and quantum discord in the case of the two field model. We conclude with a brief discussion on the wider implications of the results.