  Random number generators (RNG) based on quantum mechanics are captivating due
to their security and unpredictability compared to conventional generators,
such as pseudo-random number generators and hardware-random number generators.
This work analyzes evolutions in the extractable amount of randomness with
increasing the Hilbert space dimension, state preparation subspace, or
measurement subspace in a class of semi-device-independent quantum-RNG, where
bounding the states' overlap is the core assumption, built on the
prepare-and-measure scheme. We further discuss the effect of these factors on
the complexity and draw a conclusion on the optimal scenario. We investigate
the generic case of time-bin encoding scheme, define various input (state
preparation) and outcome (measurement) subspaces, and discuss the optimal
scenarios to obtain maximum entropy. Several input designs were experimentally
tested and analyzed for their conceivable outcome arrangements. We evaluated
their performance by considering the device's imperfections, particularly the
after-pulsing effect and dark counts of the detectors. Finally, we demonstrate
that this approach can boost the system entropy, resulting in more extractable
randomness.
