[C.A.]

I guess the easiest way to start is the website's technology page itself:

Spoiler for Tech page translation:

O.E. 311

Orbital Era 311, an era where humans start living in orbit instead of the Earth.

Time before O.E.

After exhausting resources and doing immense damage to the Earth's environment, the Earth can no longer sustain the population which has exceeded 10 billion.
Humans decided to protect and restore the Earth's environment.
But to do this, they have to move lots of people out of the Earth and live in space for a long time.
The Moon has low gravity and is not suited for long periods of living, Mars is too far to move large amounts of people effectively.
Also to colonised the Moon or Mars, you'll need to do extensive terraforming of the atmosphere, temperature, radiation shielding and other factors.
So humans instead chose to build giant space stations, colonies and live in orbit just above Earth.

Time during Space Colonisation

Space Colonies are built to support the lives of large human populations.
Space Colonies are different from Space Stations, they have their own supply of atmosphere and water, food and live supporting supplies, everything is created within the colonies themselves.
This allows humans to quickly and easily adapt to the colonies while they spend generations letting the Earth recover.
In the gigantic cylindric colonies are beautiful forests and lakes, natural environment on Earth is recreated, it makes living there almost the same as living on Earth itself.

In each Lagrange Point has a colony cluster called 'Structure', made up of a number of 'Archipelagos'.
Like the picture on the top right(refer to here) is an 'Archipelago' which consists of 7 cylinders 'Islands' connected together, the solar energy power plant Island is in the middle, the 6 other Islands are for living, industrial and city infrastructures.

1 cylinder = Island, 7 Islands = 1 Archipelago, a number of Archipelagos = Structure

Example: Surre Independent Institute = Island
Island 6 = Archipelago (There's a mess here because they use the Japanese word for 'island' to name 'Island 6, its not refering to Island as a single cylinder)
Kirkwood Structure = Structure

Akiha's school address is basically: Kirkwood Structure - Island 6 - Surre Independent Institute

Colony Location Diagram(refer to the page itself for pic)

Location each Lagrange Point:

L1: People's Republic of Uray: China and Indonesia
L2: United States of Laie: North and South America
L3: Darival Joint Sphere: All Europe and Russia
L4: Ellenburg Union: Israel, Middle East, Africa
L5: Kirkwood Co-Prosperity Sphere: Japan, Asia, Australia

The Orbitral Union, OU, consisting of representitives of L1 - L5, is the highest organisation. Followed by the governments of each Lagrange Point, then the self-governments of each Structure. Law and taxes are decided by each Lagrange Point governments, more minute decisions are made by the self-governments of each Structure.

Size Comparison

Left to Right

Island, ICP Space Station, Leopard, Akiha's ship.

Leopard
Blood Group: AB(???)
Interests: Net Surfing

The series works around QT, which actually we never got to know what it means. But basically its solely based on punning acronyms, QT is 'Cutie'. Its like telling you those girls are cuties lol, everything then works around 'QT' to continue the pun on always serious acronyms and abbreviations.

Spoiler for QT:

Q-TECTOR = Quantum Transducing Environment Crosslink Terrain Operational Reinforcer

QT-ARMS = Quantum Technology Advanced Reinforced Maneuvering Shroud.

From these 2 amazing acronyms, we know that QT is something to do with Quantum technology.

The human brain works on a Quantum level, brainwaves and all are working at the Quantum scale. In the future, humans may evolve their brains to have much stronger brainwaves and become QTs and EX-QTs.

EX-QT means Extra-QT. In Japanese, when some english word is given Extra or Over, like Extra-technology, Over-technology, it means super-technology. Extra-QT means super QT.

A QT like Akiha and Itsuki, can operate QT-ARMS. QT-ARMS movement by the QT's Quantum Brainwaves, your thoughts operate the machine. But to further enhance the control and efficiency, they put on the pilot suit, Q-TECTOR. The Q-TECTOR completely links the pilot to the QT-ARMS and gives her special gadgets and capabilities like quantumly affecting the electromagnetic spectrum to put on a disguise or become invisible etc.

The good thing about Quantum is that both humans and machines can work on his medium. So they can give machines like Leopard, Nerval and Mr Blackbelt the ability to interact at the Quantum level. Mr Blackbelt could negate the EX-QT powers of Honoka. Leopard could strip Q-TECTORS because the Q-TECTORS are basically attached on to them by some Quantum means, think mahou shoujo and quantum magic creates the Q-TECTORS.

EX-QTs all have a certain ability, Honoka and her gravity control, Episode 4's boxman creating shockwaves and Aleida gathering Tachyon particles to form the Tachyon Sword. They are also super human in physique, the school council included.

Those above are basically copypasta of my old posts in the general thread.

From a general view, the technology of this show is extremely high, especially the Brain Colonies. You can say the technology level is almost at the border of magic at certain parts, like the QT related technology and capabilities.

The Brain Colonies themselves are much higher in technology than what they have in the normal colonies. Super AI, warp, antimatter engines, time stopping freeze, inertia negation etc., its probably one of the most technologically advanced mecha anime to date.

EDIT: Just adding Aleida's Tachyon Sword.

Tachyons are hypothetical faster than light particles, definitely something out of physics. To be able to wield and control such high energy particles and not let them burst out of control, Aleida must have super high level QT powers or something.