  We examine two quantum operations, the Permutation Test and the Circle Test,
which test the identity of n quantum states. These operations naturally extend
the well-studied Swap Test on two quantum states. We first show the optimality
of the Permutation Test for any input size n as well as the optimality of the
Circle Test for three input states. In particular, when n=3, we present a
semi-classical protocol, incorporated with the Swap Test, which approximates
the Circle Test efficiently. Furthermore, we show that, with help of classical
preprocessing, a single use of the Circle Test can approximate the Permutation
Test efficiently for an arbitrary input size n.
