Current Nexus protocol is quantum resistant via a hybid SHA-3 algorithm and uses a combination of both Skein and Keccak. The 571-bit keys used are much more cryptographically secure than SHA-2 256-bit keys. And the combinatory SK algorithm increases the difficulty of any potential attack even further.
To put it in perspective, if breaking a Bitcoin key is like finding a specific grain of sand on the planet earth, then breaking a Nexus key is like filling the entire universe with sand and trying to find one specific grain somewhere in it.
That said, it is still technically only quantum resistant and not quantum proof. Eventually, a quantum computer could feasibly become powerful enough to threaten a 571-bit key, even if it takes 50 or 100 years.
Therefore, as part of the TAO framework, Nexus will take quantum resistance to the next level by releasing support for one-time use public keys that are not revealed until you use them, via signature chains. This is what the website is talking about. Here is the actual text:
Signature chains offer several advantages over equivalent quantum-resistant schemes such as BLISS and Lamport signatures, being extremely compact and lightweight, making it ideal for blockchain applications.