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The time-resolved photoconductance of amorphous and crystalline LaAlO$_3$/SrTiO$_3$ interfaces, both hosting an interfacial 2-dimensional electron gas, is investigated under irradiation by variable-wavelengths, visible or ultraviolet photons. Unlike bare SrTiO$_3$ single crystals, showing relatively small photoconductance effects, both kinds of interfaces exhibit an intense and highly persistent photoconductance with extraordinarily long characteristic times. The temporal behaviour of the extra photoinduced conductance persisting after light irradiation shows a complex dependence on interface type (whether amorphous or crystalline), sample history and irradiation wavelength. \textcolor{black}{The experimental results indicate that different mechanisms of photoexcitation are responsible for the photoconductance of crystalline and amorphous LaAlO$_3$/SrTiO$_3$ interfaces under visible light. We propose that the response of crystalline samples is mainly due to the promotion of electrons from the valence bands of both SrTiO$_3$ and LaAlO$_3$. This second channel is less relevant in amorphous LaAlO$_3$/SrTiO$_3$, where the higher density of point defects plays instead a major role., 8 pages, 5 figures, 3 tables, accepted on Scientific Reports 2015