The main aim of this paper is to establish a deep link between the totally nonnegative grassmannian and the quantum grassmannian. More precisely, under the assumption that the deformation parameter
q
q
is transcendental, we show that “quantum positroids” are completely prime ideals in the quantum grassmannian
O
q
(
G
m
n
(
F
)
)
{\mathcal O}_q(G_{mn}(\mathbb {F}))
. As a consequence, we obtain that torus-invariant prime ideals in the quantum grassmannian are generated by polynormal sequences of quantum Plücker coordinates and give a combinatorial description of these generating sets. We also give a topological description of the poset of torus-invariant prime ideals in
O
q
(
G
m
n
(
F
)
)
{\mathcal O}_q(G_{mn}(\mathbb {F}))
, and prove a version of the orbit method for torus-invariant objects. Finally, we construct separating Ore sets for all torus-invariant primes in
O
q
(
G
m
n
(
F
)
)
{\mathcal O}_q(G_{mn}(\mathbb {F}))
. The latter is the first step in the Brown-Goodearl strategy to establish the orbit method for (quantum) grassmannians.