In Voyager, the crew briefly enter
slip stream travel in two episodes. With the second removing 10 years off their voyage home.
So, how does slip stream compare to, say warp 9?
Quantum slipstream transcends the normal warp barrier by penetrating the quantum barrier with a focused quantum field. What is the quantum barrier you say? It's technobabble, and it's not explained in canon.
The net effect is that quantum slipstream technology far exceeds the speeds capable with a normal warp drive, and rivals the Borg transwarp technology. Transwarp isn't a specific type of drive, but a class of propulsion technologies that exceed the normal warp limits. In this sense, quantum slipstream would be a subset of transwarp technology.
In terms of speed comparisons, it's very hard to make a direct comparison because the writers played fast and loose with the velocities each warp factor corresponded to.
Let's start with a given: based on the episode "Hope and Fear", Voyager was able to travel 300 light years with one hour of use of the quantum slipstream drive it had.
Under the original series warp scale, traveling that distance at Warp 9 would take about 5 months.
They revised the scale for The Next Generation, and it's assumed that's what all subsequent series used. The new scale is discussed in the technical manual for The Next Generation, and is reportedly based on the formula:
On that scale, traveling that distance at Warp 9 would take about 2.3 months. This should be considered the "canon" answer, as the formula isn't explicitly contradicted by any later canon sources.
However, in "Caretaker", it's established Voyager would take 75 years at maximum speeds to reach Earth, which is 70,000 light years away. This would indicate, based on the scale for The Next Generation, Voyager's top speed to only be around Warp 7.78. This is contradicted over and over again throughout the series as Warp 9 is routinely mentioned in dialogue.
Indeed, as Memory Alpha notes, the unpublished (and consequently non-canon) technical manual for Voyager reportedly stated the "maximum speeds" talked about in "Caretaker" was actually Warp 9.6, which would indicate a different scale than that which is used in The Next Generation.
But let's discount the estimate in "Caretaker" as being too slow and continue to assume the warp scale factor hadn't changed in Voyager's time (a reasonable thing to assume since Voyager and the latter part of TNG's exploits occur simultaneously).
According to the technical manual for The Next Generation, Warp 9.6 was only sustainable on a Galaxy-class starship for no more than 12 hours. However, Voyager—an Intrepid-class starship—was designed and constructed several years after the Galaxy-class starships. We're also given every indication that Voyager is a marked improvement over previous ship designs, so it's reasonable to posit that Voyager might be indeed be able to sustain Warp 9.6 as a maximum speed, in the same vein that Janeway described it in "Caretaker".
Based on this, we can conclude that under maximum speeds, Voyager would take a little less than 2 months to reach the same distance it traveled with just one hour of quantum slipstream use.
Of course as you mentioned, in "Timeless", the claim is made that 10 years was shaved off the journey. This is where you just need to take a step back and just ignore the math. Assuming this is true, and assuming the figures for "Hope and Fear" are correct, it would mean Voyager traveled 18,900 light years.
But, wait a minute: this is crazy. "Caretaker" establishes Voyager was at most 70,000 light years from Earth, but Janeway claims they only took 10 years off their trip. Which would mean Earth was actually more than 142,000 light years away in "Caretaker".
To put that number into perspective: the diameter of the Milky Way is only 100,000 light years. Yeah, this is not going to add up.
So we're left with a paradox. To resolve it, either the numbers in "Hope and Fear" are wrong, the numbers in "Timeless" are wrong, or the quantum slipstream drive in "Timeless" is significantly slower than the one in "Hope and Fear".
For the sake of argument, let's say the one in "Timeless" is a "baby" quantum slipstream drive, and is just nowhere near the speed of the original version. Given it shaved 10 years off the trip, that means it traveled 9,333 light years. What would that tell us about Voyager's cruising speed?
Well, it tells us that it would take 10 years to travel 9,333 light years only if you were going warp 7.78, which is vastly lower than regular warp limits. I'm pretty sure they could've went faster if they got out and pushed.
And that's exactly the same estimate in "Caretaker", isn't it? Assuming Voyager's top speed isn't around warp 7 or 8, it seems Voyager is running off of a different warp scale from The Next Generation that's internally consistent, at least with these two data points. And if it is, it means Voyager was just way slower than earlier ships established in The Next Generation and Deep Space Nine.
My head's starting to hurt, but let's just continue to use The Next Generation's warp scale and compare this cruising speed to the distance mentioned in "Hope and Fear": 300 light years. To travel that distance at warp 7.78, it would take a little under 4 months.
The lesson to be learned from all of this is that the only concrete thing about quantum slipstream technology is that it's really fast and served as a fairly decent plot device to back out of a really huge and unreasonable time estimate established in the first episode.
As an aside, I point out the problems with the "Caretaker" estimate to illustrate the difficulties of direct speed comparisons, but as pointed out in the comments, it could be that Janeway really meant "maximum safe cruising speed" instead of "maximum speed", which would indicate she was unwilling to unreasonably tax Voyager's engines just to gun it back home.
In this case, the only discrepancy in warp scale factors would be the figure in the unpublished and non-canon technical manual for Voyager, and one starts to wonder if those sorts of technical discrepancies is why it was never published.
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In Caretaker, Janeway states that they are 75 years away from home at maximum warp. Since they start out about 75,000 light years from home, this presumes that Voyager's maximum warp is about 1,000 light years per year, or just under three a day. This is backed up by the episode Prime Factors, where Torres, the chief engineer says, "40,000 light years...that would knock about four decades off our trip."
Using the slipstream drive, they shave ten years off their journey, which means they traveled about ten-thousand light years in about half the length of the episode. Assuming this is in real-time, that's about five hundred light years a minute.
If you ignore the other contradictions in the series(and there are a lot of them), then slipstream velocity is thousands of times faster than warp drive. However, there are contradictions, such as:
1) In Scorpion Pt. II, Chakotay says that 40 light-years is a five-day journey at maximum warp, or 8 light-years a day. That's about three times faster than stated above, which would cut the travel time back to the Alpha Quadrant to about 22 years.
Bottom line: Slipstream velocity is much faster than warp velocity. By how much, depends on which episode you watch.
Your point is accurate. I did the calculations and I came up with the conclusion that it would have taken Voyager only 36 years, 7 months to reach Federation Space, at warp 9.6. I love the series, but the writers got the whole thing wrong. So I state, we should keep up the work to make Star Trek right. Good work guys. Keep it true. GOD's speed!