At a superconductor (S)-normal metal (N) junction pairing correlations can "leak-out" into the N region. This proximity effect [1,2] modifies the system transport properties and can lead to supercurrent flow in SNS junctions [3]. Recent experimental works showed the potential of semiconductor nanowires (NWs) as building blocks for nanometre-scale devices [4][5][6][7], also in combination with superconducting elements [8][9][10][11][12]. Here, we demonstrate an InAs NW Josephson transistor where supercurrent is controlled by hot-quasiparticle injection from normal-metal electrodes. Operational principle is based on the modification of NW electron-energy distribution [13][14][15][16][17][18][19][20] that can yield reduced dissipation and high-switching speed. We shall argue that exploitation of this principle with heterostructured semiconductor NWs opens the way to a host of out-of-equilibrium hybrid-nanodevice concepts [7,21].One practical implementation of the present InAs-NW hot-electron Josephson transistor concept is shown in Figure 1a. The SNS junction was fabricated by e-beam lithography starting from an n-doped InAs NW (the N region, for further details see Methods) and comprises two Ti/Al superconducting electrodes placed at a distance L ≃ 60 nm (S regions). Control electrodes were fabricated by depositing two additional normal-metal Ti/Au leads at the two ends of the NW. As schematically illustrated by the overlay of Fig. 1a, the N leads are used to drive a dissipative current through the NW thereby tuning Josephson coupling in the S-NW-S structure. Figure 2a shows a set of typical IV characteristics from one of the measured S-NW-S junctions at equilibrium (i.e., V inj = 0) and at different bath temperatures T . High critical currents up to I c ≃ 350 nA (corresponding to a supercurrent density ∼ 5.5 kA/cm 2 ) are observed and Josephson coupling typically persists up to about 1 K. Furthermore, from the junction differentialresistance spectra (see Supplementary Materials) we infer a superconducting order parameter ∆ = 120 µeV and a junction normal-state resistance R N ≃ 210 Ω. The I c R N product attains values as large as ≃ 75 µeV and indicates the overall success of our junction-fabrication procedure [22]. Based on carrier-density and electronmobility values (see Methods) we estimate a momentumrelaxation length ℓ ≃ 20 nm and a diffusion coefficient D = 0.02 m 2 /s. Given the junctions geometrical length L we can estimate the Thouless-energy value, i.e., the characteristic energy scale of the N region, E Th = D/L 2 ≃ 4 meV. These values indicate that our devices can be described within the frame of the diffusive short-junction limit (L > ℓ and ∆ ≪ E Th ) [22]. Figure 2b shows the evolution of I c versus T at V inj = 0 for two representative devices D1 and D2. For comparison, the theoretical prediction for an ideal short diffusive SNS junction [22,23] is also plotted assuming for both devices a supercurrent suppression of the order of 45%. Such a suppression can be expected and probably mainly stems from a...