It is great news if this algorithm can work directly on the public Internet, without requiring a specialized network - many scientific and engineering applications will be able to get its time reference directly from the Internet!
NTP and other protocols currently used are unauthenticated (there is NTP autokey, etc, but its security properties are not ideal, and mostly not deployed) and it is a big security hole, especially more and more cryptographic programs are being putting online, since this protocol is meant for financial applications, hopefully the security issues can also be solved by using digital signature.
I guess it's a rare scenario which the finance industry makes a _direct_ contribution to technology.
If I'm understanding the Huygens paper linked in one of the other comments correctly, this is strictly in-datacenter only; it relies on properties of datacenter networking that don't apply to the broader internet, in particular that routes are mostly symmetric and mostly low-latency.
Haven't read the paper yet, but even it is data-center only, it is still very relevant for science, e.g. DAQ in nuclear physics, especially if it doesn't need custom hardware.
I can't imagine how you could hope to reach nanosecond-precision over the internet without changing all the routing hardware in use today. With PTP you can already reach sub-microsecond synchronization but you need full hardware support for every network element on the route if you want to achieve that. The timestamping is done on the network PHY itself, which is how you can remove all jitter introduced by the kernel stack. Anything receiving and re-transmitting the packet along the way must update the timestamps to account for the processing delay.
Huygens is a little more clever and uses a statistical approach to sample multiple clocks and correlate them (if I understand correctly) however it still seems designed to work within a datacenter, I assume that over the internet the signal-to-noise ratio for measurements would worsen very significantly and lower the precision dramatically. It could well still outperform NTP however.
The signal-to-noise ratio would be a nuisance, naturally, but there's a more fundamental problem that's stopped anyone from trying this. Internet routing and latency is asymmetric - your request to the time server will, in general, go via a different route that takes a different amount of time compared to the message back. This introduces a bias that is indistinguishable from local clock error. Without precisely synchronised clocks at both ends, you can only determine the overall round-trip time and not the time each leg took, and therefore you cannot calculate the exact difference between your local clock and the server. This is an inhererent limit on how accurate NTP can be over the Internet. (The other reason not to bother is that GPS is cheap and can provide a very accurate synchronized clock source over long distances.)
It would probably be more accurate to install a stratum 1 NTP server on your phone since it has GPS. But then you'd probably get hit with jitter from Wi-Fi/Bluetooth/USB.
You need to get the PPS to some other device, then it could be done. The whole GPS stratum 1 thing is mostly just PPS. You get the correct time with ntp, and the start of a second with the PPS from the GPS.
NTP and other protocols currently used are unauthenticated (there is NTP autokey, etc, but its security properties are not ideal, and mostly not deployed) and it is a big security hole, especially more and more cryptographic programs are being putting online, since this protocol is meant for financial applications, hopefully the security issues can also be solved by using digital signature.
I guess it's a rare scenario which the finance industry makes a _direct_ contribution to technology.