From a production standpoint it is likely there are no relationships between the Pandorica set, the Siege Mode TARDIS and the Message Hypercubes used by the Time Lords.
If what you mean is "Could there be a relationship between the use of cubes "in-universe" relating to technologies shown in Doctor Who?" you are probably onto something, prop masters notwithstanding.
Speculation based on Observations
- Consider the humble hypercube, a mathematical construct used to describe a cube in four dimensions. Length, width, height and time.
I would imagine a species whose obsession with time and their ability to create objects using tesseracts (A TARDIS space which is bigger on the inside for example) might have an interest, nay an obsession with the cubic form, if for no other reason than it is the earliest mathematical model of extra-dimensional space.
Now imagine what we DON'T see when we look at the TARDIS or the Pandorica or a Message Hypercube.
Let's consider the Pandorica:
- A sophisticated prison designed to keep the Doctor from escaping through time.
The Pandorica was actually a prison built by the Alliance for the Doctor to stop him from inadvertently destroying all of creation in every Universe. They believed the Doctor would be responsible for the destruction of existence itself. (TV: The Pandorica Opens)
The Alliance used the lure of the Pandorica to trap The Doctor. There were many layers of security in the Pandorica including deadlocks, time stops and matter lines. It even had a restoration field to stop the Doctor from dying, which the Alliance believed a form of escape.
Around the TARDIS must flow a dimensional space capable of disrupting without leaving distortions in space time, a hyperdimensional region.
Imagine a Time Lord, being in tune enough with that hyperdimensional region to recognize where he is in time and space.
Now imagine you wanted to imprison such a being, outside of time, outside of hyper-reality space. What might that prison look like?
You got it. The fundamental structure designed to represent fourth-dimensional space, a hypercube. But because we are three dimensional beings we only see the Pandorica in three dimensions. We lack the ability to visualize the temporal locks and barriers that surround the device keeping it from being manipulated by time-travelling meddlers.
- As a result of the Pandorica's multi-dimensional existence, we cannot recognize it for what it was, a perfect prison designed to seal a being away from the hyper-reality he normally traverses at will. A prison exempt from even the TARDIS' ability to reach.
The Gallefreyan Hypercube
What about those message cubes which, coincidentally are also called hypercubes. Do you suppose it is accidental they too are in the shape of cubes? The perfect shape, indeed, the fundamental shape for a species obsessed with time travel?
- A Gallifreyan hypercube can travel through time and space and can home in on "nearby" Time Lords. They can hold psychic impressions, function as information repositories and can provide temporal coordinates. This makes them very sophisticated devices in and of themselves.
Would it matter if they were in another shape? Probably not. It's just technology. So the shape of the hypercube either helps it perform its function, or is purely ornamental (and thus showing a bit of the fetishism with time I would expect from a species like the Time Lords).
I like to think the cubic shape may have something to do with an aspect we are not privy to as three dimensional beings. Its fourth dimensional capabilities (time and space travel) may require a cubic shape in order to function. We cannot see these interactions but they may be there nonetheless.
The "Siege Mode" TARDIS
As to the "Siege Mode" TARDIS, I wasn't surprised to see the TARDIS taking on this form as it locks itself OUT of our universe as a safety precaution. (See my entry: What is the TARDIS' true weight?)
Siege mode was a feature of the Doctor's TARDIS that could be activated using a lever under the console to prevent entry or exit. Additionally, it appears to make the TARDIS impervious to all external damage. Siege mode required power to activate or deactivate.
The Twelfth Doctor activated it when his ship was in the path of an oncoming train with no shields. While in siege mode, the Doctor's TARDIS no longer had the appearance of a police box, but of a cube with Gallifreyan symbols along its sides. (TV: Flatline)
- Why the cube shape? Again, we are not privy to what fundamental aspects of reality the TARDIS is manipulating to activate its invulnerability but whatever they are, it is likely NOT just an ornamental form, given we have seen Time Lord Technology take very ordinary, sometimes even ridiculous appearances.
- Given the TARDIS is supposedly invulnerable in this form, it may be using another Gallifreyan technology, the stasis cube. Yes, when we see them, they are supposedly oil paintings containing a sliver or slice of time, but that same idea could be applied to the TARDIS when in siege mode. Frozen in a slice of time, it would be invulnerable to most conventional forces.
Gallifrey Falls No More, a stasis cube slice of time, appearing to anyone looking at it as a very sophisticated oil painting.
TARDIS Wikia regarding Stasis Cube: Later, having circumvented the time-lock and travelled back to the events of the Last Great Time War, thirteen incarnations of the Doctor came together to lock the planet Gallifrey in a stasis cube, thus saving the planet while making it seem that the Time Lords and Daleks finally obliterated each other. (TV: The Day of the Doctor)
Are the shapes of these devices, the Pandorica, the "Siege Mode" TARDIS, the message hypercubes related?
- From the production standpoint of the TV show, not at all. Convenient since they have a propmaker who already works with cubes, likes it and can make very attractive props with them.
- From a conceptual point of view: I think it could be an excellent angle to pursue showing both the underlying premise of Time Lord tesseract-driven technologies and the fundamental nature of hypercubes underlying that technology. Will anyone think of this? Probably not.