I. INTRODUCTION The theory of quantum plasma is currently rapidly expanding as a result of its potential applications ranging from micro-electronic [1] to laser-plasma interaction [2, 3] or astro- physics [4, 5]. Since the derivation of the quantum fluid equations [6, 7], many well known classical plasmas effects including plasmas oscillations with or without magnetic field [8, 9], plasma instabilities [6, 10–12], solitons and vortices [13], or turbulence [14], have been re- examined in their quantum version. Beyond their potential applications, many of these problems present a fundamental in- terest as they form the basis for the general theory of quantum plasmas. Among them, the quantum version of the two-stream instability is endorsing the rˆ ole its classical counterpart plays with classical plasma: that of a fundamental process worthy of investigation in itself, and included in every plasma physics textbook. Since the first works on the quantum two- stream instability, it has been recognized that one of the most striking difference between the quantum and the classical treatment of this process is that the quantum version displays two unstable branches instead of only one for the classical case [6]. Kinetic treatment are now available showing that thermal spread can suppress both quantum branches [15], as they do with the single unstable branch in the classical case. While unstable quantum modes in counter-streaming beams systems have been assessed for modes with wave vector aligned (two-stream) or perpendicular (filamentation [10, 11]) with the streams, intermediate orientations have not been considered yet. The intent of this paper is precisely to fill this gap. As a result, an interesting conclusion is reached: the two branches of the quantum two-stream instability are indeed connected with each other. By letting the wave vector acquire a perpendicular component, we found a continuum of unstable modes bridging the two quantum branches across the k space. We here derive the dispersion equation for arbitrarily oriented wave vector. For the sake of simplicity, we deal with the generic system formed by two cold counter-streaming electron beams of equal density n 0 / 2 and opposite velocities ± v 0 ˆ z . Ions form a fixed neutralizing background with density n$_{i}$ = n 0 and there’s no background magnetic field. The dispersion II. DISPERSION EQUATION FOR ARBITRARY MODES 2