How resilient are Proof of Stake (PoS) mechanisms to brute force quantum computing attacks? The article called "Bitcoin vs. The NSA’s Quantum Computer" was a great read pertaining to Bitcoin, and computing minimum thermodynamic energy required to brute force attack public key to discover the companion private key, Elliptic Curve Digital Signature Algorithm (ECDSA) vulnerabilities to quantum computing, and how Bitcoin public addresses hide public keys until money is spent. 

If the SHA256 and RIPE160 cryptographic hashing mechanisms are impervious to inversion (due to information collapsing loss - black hole operations) by whatever computing means used to create them, then the Bitcoin Blockchain should be fine a long as quantum computing mining does not gain much more than a 51% computing advantage for populating the Blockchain with Bitcoin's Proof of Work (PoW) mechanism. 

How will the integrity of an "energy efficient" PoS Ledger fare (during the advent of quantum computing) if it is overly reliant upon traditional digital signature technology for implementing its staking mechanism?
The main ways in which Bitcoin uses crypto algorithms are the following:


Hashing (2 x SHA256) of the block header for proof of work.
Hashing (SHA256 + RIPE160) of the public key to obtain an address (followed by base58 encoding but that's not crypto per se)
ECDSA to generate and validate signatures, allowing you spend an output.


The 2 first ways don't use ECDSA and so are not exposed to quantum computing attacks (assuming no mathematical breakthrough in prime factoring).

ECDSA signatures on the other hand may be breakable using quantum computing. So given an ECDSA public key and a quantum computer, one may be able to discover the private key.

However, public keys are only embedded in inputs when spending an output. The output itself only includes the hash of the public key, which is safe (one would need to break both ECDSA and hashing to recover the public key).

So to summarize, if a quantum computer was to be invented and assuming it could break the ECDSA asymetric encryption, the attacker could acquire all Bitcoins on addresses that have been reused (as the public key is known). Bitcoin addresses that have only been used once and proof of work should be safe. Same applies to proof of stake.