We explore the heating and differentiation of rocky planets that grow by rapid pebble accretion. Our terrestrial planets grow outside of the ice line and initially accrete 28% water ice by mass. The accretion of water stops after the protoplanet reaches a mass of 0.01 ME where the gas envelope becomes hot enough to sublimate the ice and transport the vapour back to the protoplanetary disc by recycling flows. The energy released by the decay of 26Al melts the accreted ice to form clay (phyllosilicates), oxidized iron (FeO), and a water surface layer with ten times the mass of Earth's modern oceans. The ocean- atmosphere system undergoes a run-away greenhouse effect after the effective accretion temperature crosses a threshold of around 300 K. The run-away greenhouse process vaporizes the water layer, thereby trapping the accretion heat and heating the surface to more than 6000 K. This causes the upper part of the mantle to melt and form a global magma ocean. Metal melt separates from silicate melt and sediments towards the bottom of the magma ocean; the gravitational energy released by the sedimentation leads to positive feedback where the beginning differentiation of the planet causes the whole mantle to melt and differentiate. All rocky planets thus naturally experience a magma ocean stage. We demonstrate that Earth's small excess of 182W (the decay product of 182Hf) relative to the chondrites is consistent with such rapid core formation within 5 Myr followed by equilibration of the W reservoir in Earth's mantle with 182W-poor material from the core of a planetary-mass impactor, provided that the equilibration degree is at least 25- 50%, depending on the initial Hf/W ratio. The planetary collision must have occurred at least 35 Myr after the main accretion phase of the terrestrial planets.