  We consider the spectral statistics of the Floquet operator for disordered,
periodically driven spin chains in their quantum chaotic and many-body
localized phases (MBL). The spectral statistics are characterized by the traces
of powers $t$ of the Floquet operator, and our approach hinges on the fact
that, for integer $t$ in systems with local interactions, these traces can be
re-expressed in terms of products of dual transfer matrices, each representing
a spatial slice of the system. We focus on properties of the dual transfer
matrix products as represented by a spectrum of Lyapunov exponents, which we
call \textit{spectral Lyapunov exponents}. In particular, we examine the
features of this spectrum that distinguish chaotic and MBL phases. The transfer
matrices can be block-diagonalized using time-translation symmetry, and so the
spectral Lyapunov exponents are classified according to a momentum in the time
direction. For large $t$ we argue that the leading Lyapunov exponents in each
momentum sector tend to zero in the chaotic phase, while they remain finite in
the MBL phase. These conclusions are based on results from three complementary
types of calculation. We find exact results for the chaotic phase by
considering a Floquet random quantum circuit with on-site Hilbert space
dimension $q$ in the large-$q$ limit. In the MBL phase, we show that the
spectral Lyapunov exponents remain finite by systematically analyzing models of
non-interacting systems, weakly coupled systems, and local integrals of motion.
Numerically, we compute the Lyapunov exponents for a Floquet random quantum
circuit and for the kicked Ising model in the two phases. As an additional
result, we calculate exactly the higher point spectral form factors (hpSFF) in
the large-$q$ limit, and show that the generalized Thouless time scales
logarithmically in system size for all hpSFF in the large-$q$ chaotic phase.
