  We probe the contraction from $2d$ relativistic CFTs to theories with
Bondi-Metzner-Sachs (BMS) symmetries, or equivalently Conformal Carroll
symmetries, using diagnostics of quantum chaos. Starting from an
Ultrarelativistic limit on a relativistic scalar field theory and following
through at the quantum level using an oscillator representation of states, one
can show the CFT$_2$ vacuum evolves smoothly into a BMS$_3$ vacuum in the form
of a squeezed state. Computing circuit complexity of this transmutation using
the covariance matrix approach shows clear divergences when the BMS point is
hit or equivalently when the target state becomes a boundary state. We also
find similar behaviour of the circuit complexity calculated from methods of
information geometry. Furthermore, we discuss the hamiltonian evolution of the
system and investigate Out-of-time-ordered correlators (OTOCs) and operator
growth complexity, both of which turn out to scale polynomially with time at
the BMS point.
