Fast and efficient neighbor discovery is crucial to the deployment of wireless ad hoc networks. Conventional random access neighbor discovery schemes assume a collision model for the medium access control layer and require each neighbor to transmit its identity repeatedly with random delay to resolve collision. This paper proposes a compressed neighbor discovery scheme which jointly detect all neighbors simultaneously by allowing them to simultaneously report their identity using structured signaling. The compressed neighbor discovery scheme consists of non-coherent (energy) detection followed by a simple, efficient algorithm based on group testing, which is easy to implement in wireless terminals. The performance of the scheme is characterized for networks of any given size, assuming that transmissions are subject to Rayleigh fading. The compressed neighbor discovery scheme is shown to achieve faster and more reliable neighbor discovery than existing random access schemes. For example, in a wireless ad hoc network of ten thousand nodes, where each nodes has on average six neighbors, the compressed neighbor discovery scheme is 40% faster.