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Pipeline parallelism has achieved great success in deploying large-scale transformer models in cloud environments, but has received less attention in edge environments. Unlike in cloud scenarios with high-speed and stable network interconnects, dynamic bandwidth in edge systems can degrade distributed pipeline performance. We address this issue with QuantPipe, a communication-efficient distributed edge system that introduces post-training quantization (PTQ) to compress the communicated tensors. QuantPipe uses adaptive PTQ to change bitwidths in response to bandwidth dynamics, maintaining transformer pipeline performance while incurring limited inference accuracy loss. We further improve the accuracy with a directed-search analytical clipping for integer quantization method (DS-ACIQ), which bridges the gap between estimated and real data distributions. Experimental results show that QuantPipe adapts to dynamic bandwidth to maintain pipeline performance while achieving a practical model accuracy using a wide range of quantization bitwidths, e.g., improving accuracy under 2-bit quantization by 15.85\% on ImageNet compared to naive quantization.