To improve operation parameters (S/N ratio, dynamics range, time resolution, etc.) of picosecond streak tubes, the traditional P20/P43 phosphor screens can be replaced by the others (P46/P47) having much faster decay time of the luminescence output. We provide comparative dynamic measurements of the home-made phosphor screens inside the picosecond PIF-01 streak tubes under illumination of their photocathodes by a single picosecond laser pulse or a train of picosecond laser pulses. We show that the shortest measured decay time for made-in-Russia phosphor screens (Y 3 Al 5 O 12 :Ce) is close to several hundreds of nanoseconds not only at a half-intensity level but also at a level of 10 −3 and even smaller. Furthermore, the photoelectron-to-photon conversion factor is not drastically smaller than in the traditional phosphor materials. This means that application of streak tubes supplied with fast-response phosphor screens may substantially improve the tube capabilities in the accumulation mode, and this is very important for time-resolved diagnostics of electron bunches in accelerators, where continuous accumulation of repetitive signals is needed.