What is an $\mathrm{SU}(2)$-symmetric oracle problem?

Explicit variants

1. Superpolynomial query separation

   Construct an explicitly defined invariant oracle family and a promise problem solvable with $\operatorname{poly}(n)$ STP-oracle queries but requiring superpolynomial or exponential classical queries. State how the classical oracle interface exposes the same input instance. Charge for oracle preparation, controlled access, inverses, and reference-frame resources whenever they are required.

2. Symmetric versions of known separations

   Formulate $\mathrm{SU}(2)$-symmetric analogues of Simon, forrelation, hidden-subgroup, or welded-tree problems. Preserve an explicit comparison of input size, promise, and classical/quantum access, and identify which operation supplies the separation in the invariant model.

3. Speedups lost under a symmetry restriction

   For a specified oracle separation, impose an $\mathrm{SU}(2)$ symmetry restriction on both preparation and access. Determine whether the separation survives with comparable query cost. If it fails, identify the unavailable operation and ask whether a different invariant task or primitive restores an advantage.

4. Equivariant channels and invariant instruments

   Compare a covariant channel $\mathcal E$ satisfying $$\begin{aligned}\mathcal E\!\left(U^{\otimes n}\rho(U^\dagger)^{\otimes n}\right)\\=U^{\otimes m}\mathcal E(\rho)(U^\dagger)^{\otimes m}\end{aligned}$$ with the stricter oracle whose individual Kraus operators are invariant on a fixed register. For each model, define coherent, inverse, controlled, and classical access where those notions exist. Determine which sector or multiplicity information each query can change or reveal.