Threshold signatures as a method to realize multi-party cooperation and trust distribution in blockchain have been widely studied in recent years. However, among these researches, few threshold signature schemes achieve all the properties of accountability, privacy, and key protection for the EdDSA-based blockchain systems. To fill this gap, we propose an EdDSA-based accountable threshold signature protocol with privacy and proactive refresh, named TAPS-PR. Meanwhile, we define new security models and give a detailed analysis to prove protocol security. In TAPS-PR, the threshold is variable and hidden with the signing quorum from the public view. However, the signing quorum can be traced when threshold signatures related to fraudulent events are generated. We also enhance the key security of each signer by proactive refresh, which realizes updating the private key while the public key remains unchanged. Apart from that, we present ATS-PR with increased efficiency and reduced communication cost at the cost of weaker security. The theoretical analysis and experimental results indicate that our protocols perform efficiently in terms of communication and computation overhead. Furthermore, we use Tezos, a blockchain project employing EdDSA, as a case study to demonstrate the compatibility of our protocol with real-world blockchain applications.