  This paper presents a unified approach to the modeling and computation of the
Kubo conductivity of incommensurate bilayer heterostructures at finite
temperature. Firstly, we derive an expression for the large-body limit of
Kubo-Greenwood conductivity in terms of an integral of the conductivity
function with respect to a current-current correlation measure. We then observe
that the incommensurate structure can be exploited to decompose the
current-current correlation measure into local contribution and deduce an
approximation scheme which is exponentially convergent in terms of domain size.
Secondly, we analyze the cost of computing local conductivities via Chebyshev
approximation. Our main finding is that if the inverse temperature $\beta$ is
sufficiently small compared to the inverse relaxation time $\eta$, namely
$\beta \lesssim \eta^{-1/2}$, then the dominant computational cost is
$\mathcal{O}\bigl(\eta^{-3/2}\bigr)$ inner products for a suitably truncated
Chebyshev series, which significantly improves on the
$\mathcal{O}\bigl(\eta^{-2}\bigr)$ inner products required by a naive Chebyshev
approximation. Thirdly, we propose a rational approximation scheme for the low
temperature regime $\eta^{-1/2} \lesssim \beta$, where the cost of the
polynomial method increases up to $\mathcal{O}\bigl(\beta^2\bigr),$ but the
rational scheme scales much more mildly with respect to $\beta$.
