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Schur-Weyl duality is a ubiquitous tool in quantum information. At its heartis the statement that the space of operators that commute with the tensorpowers of all unitaries is spanned by the permutations of the tensor factors.In this work, we describe a similar duality theory for tensor powers ofClifford unitaries. The Clifford group is a central object in many subfields ofquantum information, most prominently in the theory of fault-tolerance. Theduality theory has a simple and clean description in terms of finitegeometries. We demonstrate its effectiveness in several applications:  (1) We resolve an open problem in quantum property testing by showing that"stabilizerness" is efficiently testable: There is a protocol that, givenaccess to six copies of an unknown state, can determine whether it is astabilizer state, or whether it is far away from the set of stabilizer states.We give a related membership test for the Clifford group.  (2) We find that tensor powers of stabilizer states have an increasedsymmetry group. We provide corresponding de Finetti theorems, showing that thereductions of arbitrary states with this symmetry are well-approximated bymixtures of stabilizer tensor powers (in some cases, exponentially well).  (3) We show that the distance of a pure state to the set of stabilizers canbe lower-bounded in terms of the sum-negativity of its Wigner function. Thisgives a new quantitative meaning to the sum-negativity (and the related mana)-- a measure relevant to fault-tolerant quantum computation. The resultconstitutes a robust generalization of the discrete Hudson theorem.  (4) We show that complex projective designs of arbitrary order can beobtained from a finite number (independent of the number of qudits) of Cliffordorbits. To prove this result, we give explicit formulas for arbitrary momentsof random stabilizer states.

Schur-Weyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations