  In discrete-state Markovian systems, many important properties of
correlations functions and relaxation dynamics depend on the spectrum of the
rate matrix. Here we demonstrate the existence of a universal trade-off between
thermodynamic and spectral properties. We show that the entropy production
rate, the fundamental thermodynamic cost of a nonequilibrium steady state,
bounds the difference between the eigenvalues of a nonequilibrium rate matrix
and a reference equilibrium rate matrix. Using this result, we derive
thermodynamic bounds on the spectral gap, which governs autocorrelations times
and the speed of relaxation to steady state. We also derive thermodynamic
bounds on the imaginary eigenvalues, which govern the speed of oscillations. We
illustrate our approach using a simple model of biomolecular sensing.
