The ZX-calculus introduced by Bob Coecke and Ross Duncan in 2008 is a graphical language for reasoning about quantum processes using string diagrams. Quantum teleportation has an exceptionally elegant formulation in the ZX-calculus (see for instance Section 5.4 in ZX-calculus for the working computer scientist by John van de Wetering). In particular, the Bell state $\frac{1}{\sqrt{2}}(\left|00\right> + \left|11\right>)$ is represented in the ZX-calculus simply as a cup $\cup$. For me at least, this shows the ZX-calculus meets the three requirements in the question.
According to (2) on page 3 of Kindergarden quantum mechanics graduates ... or how I learned to stop gluing LEGO together and love the ZX-calculus by Bob Coecke, Dominic Horsman, Aleks Kissinger, Quanlong Wang,
'For certain quantum circuit optimisation problems, ZX-based methods now outperform the state of the art'
This is justified by a reference to Fast and effective techniques for $t$-count reduction via spider nest identities by N. de Beaudrap, X. Bian, and Q. Wang. Therefore the efficiency of the ZX-calculus as a graphical language can translate directly into efficiency (as measured by gate count) of quantum circuits.
The ZX-calculus is closely related to string diagrams for braided monoidal categories and to Penrose's string notation for tensor operations. For instance, the dual of the Bell state $\cap$ is, up to some bother with conjugation, tensor contraction. Either of these notations could merit a separate answer. For an nice example see the exercise on page 5 of Coecke's Quantum pictorialism, in which the output of a chain of four projectors on a tensor product of three Hilbert spaces is given by simple string rules.
For a textbook account see Picturing quantum processes by Bob Coecke and Aleks Kissinger. Also the website zxcalculus.com has many papers.