Continuous Group Key Agreement (CGKA) is the basis of modern Secure Group Messaging (SGM) protocols. At a high level, a CGKA protocol enables a group of users to continuously compute a shared (evolving) secret while members of the group add new members, remove other existing members, and perform state updates. The state updates allow CGKA to offer desirable security features such as forward secrecy and post-compromise security. CGKA is regarded as a practical primitive in the real-world. Indeed, there is an IETF Messaging Layer Security (MLS) working group devoted to developing a standard for SGM protocols, including the CGKA protocol at their core. Though known CGKA protocols seem to perform relatively well when considering natural sequences of performed group operations, there are no formal guarantees on their efficiency, other than the O(n) bound which can be achieved by trivial protocols, where n is the number of group numbers. In this context, we ask the following questions and provide negative answers. 1.Can we have CGKA protocols that are efficient in the worst case? We start by answering this basic question in the negative. First, we show that a natural primitive that we call Compact Key Exchange (CKE) is at the core of CGKA, and thus tightly captures CGKA’s worst-case communication cost. Intuitively, CKE requires that: first, n users non-interactively generate key pairs and broadcast their public keys, then, some other special user securely communicates to these n users a shared key. Next, we show that CKE with communication cost o(n) by the special user cannot be realized in a black-box manner from public-key encryption, thus implying the same for CGKA, where n is the corresponding number of group members.2.Can we realize one CGKA protocol that works as well as possible in all cases? Here again, we present negative evidence showing that no such protocol based on black-box use of public-key encryption exists. Specifically, we show two distributions over sequences of group operations such that no CGKA protocol obtains optimal communication costs on both sequences.