Let A be a unital C*-algebra and let σ be a one-parameter automorphism group of A. We consider QSSσ(A), the set of all quantum symmetric states on *1∞ A that are also KMS states (for a fixed inverse temperature, for specificity taken to be -1) for the free product automorphism group *1∞ σ. We characterize the elements of QSSσ(A), we show that QSSσ(A) is a Choquet simplex whenever it is nonempty and we characterize its extreme points.