  Any pure state of a qubit can be geometrically represented as a point on the
extended complex plane through stereographic projection. By employing
successive conformal maps on the extended complex plane, we can generate an
effective discrete-time evolution of the pure states of the qubit. This work
focuses on a subset of analytic maps known as fractional linear conformal maps.
We show that these maps serve as a unifying framework for a diverse range of
quantum-inspired conceivable dynamics, including (i) unitary dynamics,(ii)
non-unitary but linear dynamics and (iii) non-unitary and non-linear dynamics
where linearity (non-linearity) refers to the action of the discrete time
evolution operator on the Hilbert space. We provide a characterization of these
maps in terms of Leggett-Garg Inequality complemented with No-signaling in Time
(NSIT) and Arrow of Time (AoT) conditions.
