Time-dependent energy spectra of galactic cosmic rays (GCRs) carry fundamental information regarding their origin and propagation. When observed at the Earth, these spectra are significantly affected by the solar wind and the embedded solar magnetic field that permeates the heliosphere, changing significantly over an 11 yr solar cycle. Energy spectra of GCRs measured during different epochs of solar activity provide crucial information for a thorough understanding of solar and heliospheric phenomena. The PAMELA experiment collected data for almost 10 years (2006 June 15-2016 January 23), including the minimum phase of solar cycle 23 and the maximum phase of solar cycle 24. In this paper, we present new spectra for helium nuclei measured by the PAMELA instrument from 2010 January to 2014 September over a three-Carrington-rotation time basis. These data are compared to the PAMELA spectra measured during the previous solar minimum, providing a picture of the time dependence of the helium-nuclei fluxes over a nearly full solar cycle. Time and rigidity dependencies are observed in the proton-to-helium flux ratios. The force-field approximation of the solar modulation was used to relate these dependencies to the shapes of the local interstellar proton and helium-nuclei spectra.