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We provide quantitative bounds on the characterisation of multiparticleseparable states by states that have locally symmetric extensions. The boundsare derived from two-particle bounds and relate to recent studies on quantumversions of de Finetti's theorem. We discuss algorithmic applications of ourresults, in particular a quasipolynomial-time algorithm to decide whether amultiparticle quantum state is separable or entangled (for constant number ofparticles and constant error in the LOCC or Frobenius norm). Our resultsprovide a theoretical justification for the use of the Search for SymmetricExtensions as a practical test for multiparticle entanglement.

Detection of Multiparticle Entanglement: Quantifying the Search for Symmetric Extensions