Deep inside planets, extreme density, pressure, and temperature strongly modify the properties of the constituent materials. In particular, how much heat solids can sustain before melting under pressure is key to determining a planet's internal structure and evolution. We report laser-driven shock experiments on fused silica, alpha-quartz, and stishovite yielding equation-of-state and electronic conductivity data at unprecedented conditions and showing that the melting temperature of SiO2 rises to 8300 K at a pressure of 500 gigapascals, comparable to the core-mantle boundary conditions for a 5-Earth mass super-Earth. We show that mantle silicates and core metal have comparable melting temperatures above 500 to 700 gigapascals, which could favor long-lived magma oceans for large terrestrial planets with implications for planetary magnetic-field generation in silicate magma layers deep inside such planets.