The growth of supermassive black holes (BHs) located at the centers of their host galaxies comes mainly from the accretion of gas, but how to fuel them remains an outstanding unsolved problem in quasar evolution. This issue can be elucidated by quantifying the radiative efficiency parameter (η) as a function of redshift, which also provides constraints on the average spin of the BHs and its possible evolution with time. We derive a formalism to link η with the luminosity density, BH mass density, and duty cycle of quasars, quantities we can estimate from existing quasars, and galaxy survey data. We find that η has a strong cosmological evolution: at z≈ 2, η≈ 0.3, and by z≈ 0 it has decreased by an order of magnitude, to η≈ 0.03. We interpret this trend as evolution in BH spin, and we appeal to episodic, random accretion as the mechanism for reducing the spin. The observation that the fraction of radio-loud quasars decreases with increasing redshift is inconsistent with the popular notion that BH spin is a critical factor for generating strong radio jets. In agreement with previous studies, we show that the derived history of BH accretion closely follows the cosmic history of star formation, consistent with other evidence that BHs and their host galaxies co-evolve.