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Pseudorandom quantum states (PRSs) and pseudorandom unitaries (PRUs) possessthe dual nature of being efficiently constructible while appearing completelyrandom to any efficient quantum algorithm. In this study, we establishfundamental bounds on pseudorandomness. We show that PRSs and PRUs exist onlywhen the probability that an error occurs is negligible, ruling out theirgeneration on noisy intermediate-scale and early fault-tolerant quantumcomputers. Additionally, we derive lower bounds on the imaginarity andcoherence of PRSs and PRUs, rule out the existence of sparse or real PRUs, andshow that PRUs are more difficult to generate than PRSs. We introducepseudoresource, where states of with low amount of a given resource masqueradeas high-resource states. We define pseudocoherence, pseudopurity andpseudoimaginarity, and identify three distinct types of pseudoresources interms of their masquerading capabilities. Our work also establishes rigorousbounds on the efficiency of property testing, demonstrating the exponentialcomplexity in distinguishing real quantum states from imaginary ones, incontrast to the efficient measurability of unitary imaginarity. Lastly, we showthat the transformation from a complex to a real model of quantum computationis inefficient, in contrast to the reverse process, which is efficient. Ourresults establish fundamental limits on property testing and provide valuableinsights into quantum pseudorandomness.

Pseudorandom unitaries are neither real nor sparse nor noise-robust