  We study the connections between three quantities that can be used as
diagnostics for quantum chaos, i.e., the out-of-time-order correlator (OTOC),
Loschmidt echo (LE), and complexity. We generalize the connection between OTOC
and LE for infinite dimensions and extend it for higher-order OTOCs and
multi-fold LEs. Novel applications of this intrinsic relation are proposed. We
also propose a relationship between a specific circuit complexity and LE by
using the inverted oscillator model. These relationships signal a deeper
connection between these three probes of quantum chaos.
