Earlier quoted context omitted.
Or stateless verification when "other people" are allowed to create claims. There are certainly federation scenarios where PKI alone could be sufficient (as opposed to complex handshakes such as OpenID Connect or SAML). Obviously in accepting those claims you get to treat every claim with a giant grain of salt, but there are auth models where that makes sense (the only claim I need to trust is the certificate itself…
I’m not sure I follow: if the other party is allowed to create claims too, symmetric crypto seems more obvious. That’s what NS (and later KRBv5) did in the seventies. Or is that precisely what you’re saying?
I've long considered the merits of a kerberos-like system built on top of something like nacl... But without out-of-the-box support from all kinds of systems... It'd essentially build down to ssh+certificates with expiration dates... So i've gathered "better CA for ssh" is the better product. And there are thankfully a couple of projects in that vein (teleport, netflix/bless, others?).
[ed: i should add: I think tptacek is absolutely right about public key systems being easier to get wrong; but part of that is also the problem domain: look at the history of security issues with Kerberos (both implementations and protocol evolution) for a great example. On the face of it NxN key exchange is "text book simple; should be easy to define, a little tricky to scale". Then there's replay, clock drift, (de)serialisation, nounces, large number of session keys (secure random numbers)...]