Jan Nöller

Learning properties of quantum states from measurement data is a fundamental challenge in quantum information. The sample complexity of such tasks depends crucially on the measurement primitive. While shadow tomography achieves sample- efficient learning by allowing entangling measurements across many copies, it requires prohibitively deep circuits. At the other extreme, two-copy measurements already yield exponential advantages over single-copy strategies in tasks such as Pauli tomography. In this work we show that such sharp separations extend far beyond the two-copy regime: for every prime k we construct explicit learning tasks of degree k, which are exponentially hard with (k − 1)-copy measurements but efficiently solvable with k- copy measurements. Our protocols are not only sample-efficient but also realizable with shallow circuits. Extending further, we show that such finite-degree tasks ex- ist for all square-free integers k, pointing toward a general principle underlying their existence. Together, our results reveal an infinite hierarchy of multi-copy learning prob- lems, uncovering new phase transitions in sample complexity and underscoring the role of reliable quantum memory as a key resource for exponential quantum advantage