The Great Firewall of China is controlled by the government and has put in place a "learning algorithm" to block unapproved encrypted traffic. The only way to get encrypted traffic through the firewall is to get it approved from the government.
Here is a link for more information.
What sort of approaches are already out there to circumvent this?
Are there any theoretical approaches that can circumvent this?
You might use low-bandwidth sidechannels. For example, one could set up a site offering pictures, and the site could sport a "thumbnail wall". HTTP protocol allows for requesting those thumbnails in any order, which means that using twenty pictures you can encode up to 20! (i.e., around 60 bits) in the request scheme. There is no way of detecting or blocking such a scheme short of computationally unfeasible statistical analyses, proxying with request order randomization, or HUMINT of course.
A more easily detectable scheme, with more bandwidth, allows encoding information in ETag and Proxy headers.
Then, one could try more cloak-and-dagger schemes: for example one could bounce ICMP packets with faked source address against a trusted (by the government) host. This kind of malicious activity is apparently not (yet) deemed dangerous, and several firewalled hosts will blissfully bounce back the ICMP packet towards the apparent source, thereby penetrating the firewall.
Then again, incredibly, the Great Firewall of China doesn't seem to care about spam. You'd be surprised at the quantity of unsolicited commercial emails, often containing bulky images, reaching the rest of the world from China. Probably, any sufficiently asymmetric traffic towards port 25/tcp is considered symptomatic of the presence of a Chinese criminal, and the firewall has other priorities. Thanks to some helpful spammers in Guangdong, I can confirm that sending back up to 1 Kb encrypted, compressed and base64-encoded error messages has no effect whatsoever, and the incoming connection keeps requesting relay for more and more spam, which is accepted (and discarded). This channel appears to be capable of about 64 kbit bidirectional.
All this requires encapsulating VPN traffic into different protocols, similar to what Haystack did. Other solutions exist - e.g. FreeGate -, but of course the more widely known they are, the faster Golden Shield is going to (try to) bust them. Simpler strategies aimed at defeating a firewall could just entail obfuscating an existing protocol using one-time pad and supplying rapidly changing external endpoints (that's what the ultrasurf utility did), and betting that you can come up with more obfuscations, and quicker, than the GFC people can come up with fixes.
The risk there, though, is that the GFC switch from "blacklist" to "whitelist" operation: first mowing down the traffic to manageable sizes, then actively rewriting rather than rerouting all known protocols, enforcing strong grammar checks and anomaly detection. For example, refusing HTTPS and TLS unless you accept a nonmatching, GFC-generated certificate to allow man-in-the-middle SSL decryption (claiming that this "does not limit responsible citizens", unless "they have something to hide"), and otherwise refusing all packets that it does not fully understand. This is more or less an answer to Ai Weiwei's statement, that the only way to control the Internet is to shut it down. And the answer would be "Well then, how about we do just that".
Steganography was designed for exactly such scenarios. More broadly, look for concept of plausible deniability that helps people to cope with dangers of unaccounted activities in hostile environments.
Of course, in China's scenario, a dedicated service must be established in outer world to support steganography-enabled communication with Chinese oppressed network. I don't suppose it as problem though, as usually there's abundance of volunteering people supporting values of liberty in communication.
I've actually very recently had to do that. We have a set of servers in Beijing that must be able to communicate with our US offices securely (mostly for sending logs and audit data). We use openvpn to do it in all other geolocations, but we have quickly discovered that openvpn traffic is rapidly recognized and blocked in China -- because openvpn handshake has telltale signs.
The somewhat ridiculous solution to this is to wrap openvpn traffic in stunnel. I can confirm that it works and hasn't been blocked so far. It's clearly not a great solution, since on top of the latency of all connections out of China, you're adding the latency of openvpn over tcp over tcp again, twice-encrypted, but at least it works, as opposed to not at all.
This "learning algorithm" probably means that the GFC first detects encrypted traffic which might be a VPN or Tor and then reconnects to the communication destination and tries to "speak" VPN or Tor to verify this suspicion. This active probing is confirmed to happen for the Tor network (details are here) and the technique might be similar if not the same for the recent block of VPN connections.
Now there is a lot of exciting research in the field of blocking-resistant communication. On the one hand, people are working on transport protocols which look like "randomness". Examples are obfs2 and Dust. Other circumvention protocols mimic existing protocols and would, as a result, survive a censor which chooses to whitelist the Internet. Examples are Code Talker Tunnel or StegoTorus. Mimicing other protocols usually provides stronger resistance but reduced throughput due to the steganographic overhead. After all, there are many censors which act highly different. The GFC is just one example. Therefore, we need many different circumvention protocols to tackle all the different censorship techniques.
While it will always be possible to transmit a couple of bits in a covert fashion, the challenge lies in low-latency systems which allow comfortable web surfing. That's what people want, after all. An overview of censorship-related security research is available here.
Off the top of my head, I can think of many ways to beat China's Great Wall, and some of these things may only be useful just once and they border on espionage.
