In the first part of the play, Polixenes, King of Bohemia, is the guest of Leontes, King of Sicilia,...
Shakespeare's The Winter’s Tale, a complex play which contains elements of tragedy, comedy, and romance, unfolds in two parts, with acts 1–3 occurring sixteen years before acts 4 and 5.
In the first part of the play, Polixenes, King of Bohemia, is the guest of Leontes, King of Sicilia, his friend since boyhood. When Polixenes decides to return to Bohemia at the beginning of act 1, scene 2, Leontes urges him to stay longer in Sicilia, and asks his wife, Hermione, to persuade Polixenes to extend his visit with them.
Within a very short period of time—barely thirty-five lines, or about two minutes of stage time—Hermione persuades Polixenes to stay in Sicilia.
Almost immediately, Leontes becomes suspicious and jealous of Hermione and Polixenes. Leontes decides, based on no evidence whatsoever, that Polixenes and Hermione are secret lovers.
Leontes enlists his friend, Camillo, to poison Polixenes. Camillo thinks better of his arrangement with Leontes to kill Polixenes, and rather than poison Polixenes (and quickly end the play), Camillo risks his own life by revealing Leontes's plan to Polixenes, whereupon Camillo and Polixenes decide that discretion is the better part of valor and leave Bohemia as fast as they can.
Aside from the speed with which Hermione convinces Polixenes to stay in Bohemia—almost as if there was already a plan in place between Polixenes and Hermione for Polixenes to stay—there is no evidence that Polixenes and Hermione have any kind of secret, amorous relationship.
Leontes's jealousy of Polixenes and his belief in Hermione's infidelity appears completely irrational. If Polixenes and Hermione previously conspired to have Polixenes stay in Bohemia for their own purposes, the issue would never have arisen with Leontes. Polixenes would not have said that he was leaving Bohemia, and he would simply had stayed in Bohemia, and with Hermione, without being asked by Leontes or persuaded by Hermione to do so.
No other character in the play has any idea or gives any indication that Leontes has the emotional capacity for the level of jealousy that he exhibits towards Polixenes and Hermione. Leontes gives no indication in the early part of the play that he's at all inclined towards jealousy or that he's even beginning to have any jealous feelings towards Polixenes until his jealousy erupts, catching everyone by surprise.
Also, Leontes has been married to Hermione for several years, and he hasn't previously demonstrated any jealousy towards her or any other of their shared male friends. Additionally, Polixenes has been staying with Leontes and Hermione for nine months when Leontes's jealousy suddenly manifests itself.
As a measure of Leontes's irrational frame of mind, it took far longer and required much more manipulation, insinuation, innuendo, and staged scenarios on Iago's part to inflame Othello's jealousy in Othello to the same level of murderous rage that Leontes's jealousy reaches in just a few minutes. In all, Leontes's frighteningly swift, intense, and all-consuming jealousy simply defies rational explanation.