  In this paper, we present a rigorous proof of the convergence of first order
and second order exponential time differencing (ETD) schemes for solving the
nonlocal Cahn-Hilliard (NCH) equation. The spatial discretization employs the
Fourier spectral collocation method, while the time discretization is
implemented using ETD-based multistep schemes. The absence of a higher-order
diffusion term in the NCH equation poses a significant challenge to its
convergence analysis. To tackle this, we introduce new error decomposition
formulas and employ the higher-order consistency analysis. These techniques
enable us to establish the $\ell^\infty$ bound of numerical solutions under
some natural constraints. By treating the numerical solution as a perturbation
of the exact solution, we derive optimal convergence rates in
$\ell^\infty(0,T;H_h^{-1})\cap \ell^2(0,T; \ell^2)$. We conduct several
numerical experiments to validate the accuracy and efficiency of the proposed
schemes, including convergence tests and the observation of long-term
coarsening dynamics.
