Shape sensing can be accomplished using optical fiber sensors through different interrogation principles such as fiber Bragg gratings, optical frequency-domain reflectometry (OFDR), or optical time-domain reflectometry (OTDR). These techniques are either not entirely distributed, have poor performance in dynamic sensing, or are only valid for few-meter-long fibers. Here, we present a system able to perform distributed curvature sensing with a range of 125 m, 10-cm resolution, and a sampling rate of 50 Hz. This is done by interrogating three cores of a multi-core fiber (MCF) with the novel, to the best of our knowledge, time-expanded phase-sensitive (TE-Φ)OTDR technique. This system fills a performance gap in fiber shape sensors, opening the door to applications in civil engineering, medicine, or seismology.