  Recent open quantum system studies showed that quarkonium time evolution
inside the quark-gluon plasma is determined by transport coefficients that are
defined in terms of a gauge invariant correlator of two chromoelectric field
operators connected by an adjoint Wilson line. We study the Euclidean version
of the correlator for quarkonium evolution and discuss the extraction of the
transport coefficients from this Euclidean correlator, highlighting its
difference from other problems that also require reconstructing a spectral
function, such as the calculation of the heavy quark diffusion coefficient.
Along the way, we explain why the transport coefficient $\gamma_{\rm adj}$
differs from $\gamma_{\rm fund}$ at finite temperature at $\mathcal{O}(g^4)$,
in spite of the fact that their corresponding spectral functions differ only by
a temperature-independent term at the same order. We then discuss how to
evaluate the Euclidean correlator via lattice QCD methods, with a focus on
reducing the uncertainty caused by infrared renormalons in determining the
renormalization factor nonperturbatively.
