Hash-based asymmetrical encryption (not digital signature) schemes?
It's impossible to construct a strong asymmetric encryption algorithm that relies only on the security of a hash function:


  We prove that every key exchange protocol in the random oracle model in which the honest users make at most n queries to the oracle can be broken by an adversary making O(n^2) queries to the oracle.


Boaz Barak, Mohammad Mahmoody-Ghidary - Merkle Puzzles are Optimal

(The proof assumes the attacker is only limited by the number of hash computations and has unlimited computational power otherwise)

The quadratic advantage of Merkle Puzzles is not sufficient to achieve the security and performance levels expected of modern cryptography. For example at a security level of 80 bits (Breaking it is equivalent to about one day of bitcoin mining), the defenders would need to exchange terabytes of information and need an hour or so of computation.

And against quantum computers you don't even get that quadratic advantage.