  Quantum communication with systems of dimension larger than two provides
advantages in information processing tasks. Examples include higher rates of
key distribution and random number generation. The main disadvantage of using
such multi-dimensional quantum systems is the increased complexity of the
experimental setup. Here, we analyze a not-so-obvious problem: the relation
between randomness certification and computational requirements of the
postprocessing of experimental data. In particular, we consider semi-device
independent randomness certification from an experiment using a four
dimensional quantum system to violate the classical bound of a random access
code. Using state-of-the-art techniques, a smaller quantum violation requires
more computational power to demonstrate randomness, which at some point becomes
impossible with today's computers although the randomness is (probably) still
there. We show that by dedicating more input settings of the experiment to
randomness certification, then by more computational postprocessing of the
experimental data which corresponds to a quantum violation, one may increase
the amount of certified randomness. Furthermore, we introduce a method that
significantly lowers the computational complexity of randomness certification.
Our results show how more randomness can be generated without altering the
hardware and indicate a path for future semi-device independent protocols to
follow.
