Abstract: Two-message quantum key exchange (2QKE) cannot be realized in a statistically secure way, and therefore some computational assumptions are required. So far, the known weakest assumption for 2QKE is the existence of one-way functions (OWFs). Can we construct 2QKE from a weaker assumption than OWFs? In this paper, we give the first complete characterization of 2QKE. We show that the following four primitives are (existentially) equivalent: everlasting-secure 2QKE, standard 2QKE, efficiently-verifiable state puzzles (EV-StatePuzzs), and one-time secure digital signatures with quantum signatures and signing keys. Because EV-StatePuzzs are potentially weaker than OWFs [Kretschmer, Qian and Tal 2025], the equivalence implies that 2QKE are potentially weaker than OWFs. Moreover, our result shows the equivalence between standard 2QKE and everlasting-secure 2QKE, which means that everlasting security comes for free. Finally, one-shot signatures, quantum lightning, and public-key quantum money mini-schemes imply EV-StatePuzzs, and therefore, our result also gives a new lowerbound, namely everlasting-secure 2QKE, of these public-key unclonable primitives. We also show that one-shot signatures imply QPKE with quantum public keys. [Kitagawa, Morimae,Nishimaki and Yamakawa, 2024] constructed QPKE with quantum public keys from OWFs, and it was open whether QPKE with quantum public keys can be constructed from an assumption weaker than or incomparable to OWFs. Our result gives a potential answer to this question since one-shot signatures seem to be incomparable with OWFs. Moreover, QPKE constructed from one-shot signatures has an additional functionality, which we call certified freeze, that allows the receiver to publish a publicly verifiable classical certificate that computationally binds the receiver to an upper bound on the total number of distinct valid ciphertexts under its verification key.