  Electrical and thermal conductances of a quantum Hall bar reflect the
topological structure of the incompressible bulk phase. Here we show that noise
of electrical current carried through the edge evidences the interplay between
these two topological observables. Transport through a structured edge is
modeled by a voltage-biased line junction made up of two counter-propagating
modes associated with respective filling factors. Specifically, we focus on the
edge of a $\nu=2/3$ fractional quantum Hall state. Noise is generated at a
point distinctly separated from the hot spot (where most of the Ohmic
dissipation takes place) and reflects the competition between ballistically
carried downstream current and diffusively carried heat (which can propagate
also upstream). We propose specific setups where our predictions can be
measured.
