Classical shadows are succinct classical representations of quantum states which allow one to encode a set of properties P of a quantum state ρ, while only requiring measurements on logarithmically many copies of ρ in the size of P. In this work, we initiate the study of verification of classical shadows, denoted classical shadow validity (CSV), from the perspective of computational complexity, which asks: Given a classical shadow S, how hard is it to verify that S predicts the measurement statistics of a quantum state? We first show that even for the elegantly simple classical shadow protocol of [Huang, Kueng, Preskill, Nature Physics 2020] utilizing local Clifford measurements, CSV is QMA-complete. This hardness continues to hold for the high-dimensional extension of said protocol due to [Mao, Yi, and Zhu, PRL 2025]. In contrast, we show that for the HKP and MYZ protocols utilizing global Clifford measurements, CSV can be "dequantized" for low-Frobenius norm observables, i.e., solved in randomized poly-time with standard sampling assumptions. Finally, we show that CSV for exponentially many observables is complete for a quantum generalization of the second level of the polynomial hierarchy, yielding the first natural complete problem for such a class.

Keyword-based heat maps are a natural way to explore and analyze the spatial properties of social media data. Dealing with large datasets, there may be many different keywords, making offline pre-computations very hard. Interactive frameworks that exploit database sampling can address this challenge. We present a novel middleware technique called Heatflip, which issues diametrically opposed samples into the temporal and spatial dimensions of the data stored in an external database. Spatial samples provide insights into the temporal distribution and vice versa. The progressive exploration approach benefits from adaptive indexing and combines the retrieval and visualization of the data in a middleware layer. Without any a priori knowledge of the underlying data, the middleware can generate accurate heat maps in 85% shorter processing times than conventional systems. In this paper, we discuss the analytical background of Heatflip, showcase its scalability, and validate its performance when visualizing large amounts of social media data.