AbstractTo determine the topological quantum numbers of fractional quantum Hall (FQH) states hosting counter-propagating (CP) downstream (Nd) and upstream (Nu) edge modes, it is pivotal to study quantized transport both in the presence and absence of edge mode equilibration. While reaching the non-equilibrated regime is challenging for charge transport, we target here the thermal Hall conductance GQ, which is purely governed by edge quantum numbers Nd and Nu. Our experimental setup is realized with a hexagonal boron nitride (hBN) encapsulated graphite gated single layer graphene device. For temperatures up to 35 mK, our measured GQ at ν = 2/3 and 3/5 (with CP modes) match the quantized values of non-equilibrated regime (Nd + Nu)κ0T, where κ0T is a quanta of GQ. With increasing temperature, GQ decreases and eventually takes the value of the equilibrated regime ∣Nd − Nu∣κ0T. By contrast, at ν = 1/3 and 2/5 (without CP modes), GQ remains robustly quantized at Ndκ0T independent of the temperature. Thus, measuring the quantized values of GQ in two regimes, we determine the edge quantum numbers, which opens a new route for finding the topological order of exotic non-Abelian FQH states.