Constant-accuracy estimation and full counting hardness
Determine whether estimating stabilizer Rényi entropies of two-dimensional projected entangled pair states to constant additive accuracy retains the $$-hardness of exact stabilizer-entropy evaluation, rather than only the $=$-hardness established in the paper.
References
An open question is whether constant-accuracy estimation also retains the $$-hardness of exact SE evaluation, beyond the $=$-hardness established here.
Several other directions remain open. It would be important to understand how the present hardness changes under additional physical constraints, such as injectivity, translational invariance, the existence of a uniformly gapped parent Hamiltonian and its possible symmetries. Since the reductions already use constant physical dimension and constant bond dimension, any efficient subclass must rely on structure beyond bounded local dimensions. Further extensions include other two-dimensional Ans"atze such as isometric TN, and other nonstabilizerness monotones or stabilizer-based witnesses. Clarifying which of these quantities remain hard, and which become tractable under physically natural restrictions, would sharpen the role of nonstabilizerness in tensor-network simulation of quantum many-body systems.