In the video game Final Fantasy, there's a spell called "Doom" which places a countdown over your head. When it reaches 0, you die.
RSA is Doomed in exactly that sense: it's just a matter of time before RSA offers zero security, whereas elliptic curve cryptography remains (for now) unbroken.
Whatever reason to prefer elliptic curves over integer factorization or discrete log-based schemes must be classical.
Then which (if any) algorithm is currently believed to be safe from quantum computing?
I think I was thinking of lattice-based cryptography.
You didn't ask, but it's worth saying: block ciphers, stream ciphers and hash functions aren't thought to be fundamentally threatened by QC the way IFP and DLP number theoretic cryptosystems are.
A switch to lattice or code-based crypto seems unlikely in the near future.
This is not the only possible outcome. Scalable quantum computing could turn out to be impractical due to cost. Or new physics could be found that rules out scalable quantum computing. Personally, I find both of these outcomes unlikely, the second more than the first, but they do exist.