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New short-pulse kilojoule, Petawatt-class lasers, which have recently come online and are coupled to large-scale, many-beam long-pulse facilities, undoubtedly serve as very exciting tools to capture transformational science opportunities in high energy density physics. These shortpulse lasers also happen to reside in a unique laser regime: very high-energy (kilojoule), relatively long (multi-picosecond) pulse-lengths, and large (10s of micron) focal spots, where their use in driving energetic particle beams is largely unexplored. Proton acceleration via Target Normal Sheath Acceleration (TNSA) using the Advanced Radiographic Capability (ARC) short-pulse laser at the National Ignition Facility in the Lawrence Livermore National Laboratory is demonstrated for the first time, and protons of up to 18 MeV are measured using laser irradiation of >1 ps pulse-lengths and quasi-relativistic ($10 18 W/cm 2 ) intensities. This is indicative of a super-ponderomotive electron acceleration mechanism that sustains acceleration over long (multi-picosecond) time-scales and allows for proton energies to be achieved far beyond what the well-established scalings of proton acceleration via TNSA would predict at these modest intensities. Furthermore, the characteristics of the ARC laser (large $100 lm diameter focal spot, flat spatial profile, multi-picosecond, relatively low prepulse) provide acceleration conditions that allow for the investigation of 1D-like particle acceleration. A high flux $ 50 J of laser-accelerated protons is experimentally demonstrated. A new capability in multi-picosecond particle-in-cell simulation is applied to model the data, corroborating the high proton energies and elucidating the physics of multi-picosecond particle acceleration.