Arbiters are the essential components of the Network-On-Chip (NOC) systems and are used to resolve the contention problem where multiple requests must be handled for shared resources. On the other hand, with the ever-increasing downsizing trend in the fabrication technology, Quantum-dot Cellular Automata (QCA) with its nano scales and very low power consumption is a promising candidate for implementing future NOCs. In the current work, we design and simulate nano-arbiters using QCA with the following contributions: i) The 2-bit Basic Round Robin Arbiter (RRA) and the 2-bit Ping Pong Arbiter (PPA) are designed and simulated; ii) A solution for an erroneous condition found in the original circuit of RRA is reported and fixed; iii) We use Cartesian Genetic Programming (CGP) approach to simplify the RRA and PPA designs; iv) In order to leverage our QCA designs, we apply a more realistic clock distribution (2-DW clocking) and report the results. At the end, a one-to-one comparison of the two arbiters designed with QCA will be presented using such benchmarks as area, latency, etc. Our results show that in the 2-bit input mode, the PPA arbiter has the best overall performance.