  We prove lower bound for the first closed or Neumann nonzero eigenvalue of
the Laplacian on a compact quaternion-K\"ahler manifold in terms of dimension,
diameter, and scalar curvature lower bound. It is derived as large time
implication of the modulus of continuity estimates for solutions of the heat
equation. We also establish lower bound for the first Dirichlet eigenvalue in
terms of geometric data, via a Laplace comparison theorem for the distance to
the boundary function.
