  Pseudorandom quantum states (PRSs) and pseudorandom unitaries (PRUs) possess
the dual nature of being efficiently constructible while appearing completely
random to any efficient quantum algorithm. In this study, we establish
fundamental bounds on pseudorandomness. We show that PRSs and PRUs exist only
when the probability that an error occurs is negligible, ruling out their
generation on noisy intermediate-scale and early fault-tolerant quantum
computers. Further, we show that PRUs need imaginarity while PRS do not have
this restriction. This implies that quantum randomness requires in general a
complex-valued formalism of quantum mechanics, while for random quantum states
real numbers suffice. Additionally, we derive lower bounds on the coherence of
PRSs and PRUs, ruling out the existence of sparse PRUs and PRSs. We also show
that the notions of PRS, PRUs and pseudorandom scramblers (PRSSs) are distinct
in terms of resource requirements. We introduce the concept of pseudoresources,
where states which contain a low amount of a given resource masquerade as
high-resource states. We define pseudocoherence, pseudopurity and
pseudoimaginarity, and identify three distinct types of pseudoresources in
terms of their masquerading capabilities. Our work also establishes rigorous
bounds on the efficiency of property testing, demonstrating the exponential
complexity in distinguishing real quantum states from imaginary ones, in
contrast to the efficient measurability of unitary imaginarity. Lastly, we show
that the transformation from a complex to a real model of quantum computation
is inefficient, in contrast to the reverse process, which is efficient. Our
results establish fundamental limits on property testing and provide valuable
insights into quantum pseudorandomness.
