  We make lattice generalization of two well-known zero dimensional models of
quantum spin glass, Sachdev-Ye (SY) and spherical quantum $p$-spin glass model,
to one dimension for studying crossovers in non-local scrambling dynamics due
to glass transition, complex dynamics, and quantum and thermal fluctuations in
paramagnetic (PM) and spin glass (SG) phases. In the SY chain of quantum dots,
each described by infinite-range random Heisenberg model of $N$ spin-$S$
$SU(M)$ spins, we obtain the quantum Lyapunov exponent $\lambda_\mathrm{L}$ and
butterfly velocity $v_B$ as a function of temperature $T$ and the quantum
parameter $S$ across the PM-SG phase boundary using a bosonic spinon
representation in the large-$N,M$ limit. In particular, we extract asymptotic
$T$ and $S$ dependence, e.g., power laws, for $\lambda_\mathrm{L}$ and $v_B$ in
different regions deep inside the phases and near the replica symmetry breaking
SG transition. We find the chaos to be non-maximal almost over the entire phase
diagram. Very similar results for chaos indicators are found for the $p$-spin
glass chain as a function of temperature and a suitable quantum parameter
$\Gamma$, with some important qualitative differences. In particular,
$\lambda_\mathrm{L}$ and $v_B$ exhibit a maximum, coinciding with onset of
complex glassy relaxation, above the glass transition as a function of $T$ and
$\Gamma$ in the PM phase of the $p$-spin glass model. In contrast, the maximum
is only observed as a function of $S$, but not with temperature, in the PM
phase of SY model. The maximum originates from enhanced chaos due to maximal
complexity in the glassy landscape. Thus, the results in the SY model indicate
very different evolution of glassy complexity with quantum and thermal
fluctuations.
